Universal KZB equations: the elliptic case

Damien Calaque^{1}, Benjamin Enriquez^{2}, and Pavel Etingof^{3}

1 Université Lyon 1, Institut Camille Jordan (CNRS), F-69622 Villeurbanne,

  France calaque@math.univ-lyon1.fr

2 ULP Strasbourg 1 et IRMA (CNRS), 7 rue René Descartes, F-67084 Strasbourg,

  France enriquez@math.u-strasbg.fr

3 Department of Mathematics, MIT, Cambridge, MA 02139, USA

  etingof@math.mit.edu

To Yuri Ivanovich Manin on his 70th birthday

Summary. We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the 𝔰l_{n} case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant Dmodules on 𝔰l_{n} to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over 𝔰l_{n} which are supported on the nilpotent cone.

Introduction

The KZ system was introduced in [KZ84] as a system of equations satisfied by correlation functions in conformal field theory. It was then realized that this system has a universal version [Dri91]. The monodromy of this system leads to representations of the braid groups, which can be used for proving the that the pure braid groups, which are the fundamental groups of the configuration spaces of ℂ, are formal (i.e., their Lie algebras are isomorphic with their associated graded Lie algebras, which is a holonomy Lie algebra and thus has an explicit presentation). This fact was first proved in the framework of minimal model theory [Sul77, Koh83]. These results gave rise to Drinfeld’s theory of associators and quasi-Hopf algebras [Dri90b, Dri91]; one of the purposes of this work was to give an algebraic construction of the formality isomorphisms, and indeed one of its by-products is the fact that these isomorphisms can be defined over ℚ.

In the case of configuration spaces over surfaces of genus ≥ 1, similar Lie algebra isomorphisms were constructed by Bezrukavnikov [Bez94], using results of Kriz [Kri94]. In this series of papers, we will show that this result can be reproved using a suitable flat connection over configuration spaces. This connection is a universal version of the KZB connection [Ber98a, Ber98b], which is the higher genus analogue of the KZ connection.

In this paper, we focus on the case of genus 1. We define the universal KZB connection (Section 1), and rederive from there the formality result (Section 2). As in the integrable case of the KZB connection, the universal KZB connection extends from the configuration spaces C̄(E_{τ} , n)/S_{n} to the moduli space M_{1,[n]} of elliptic curves with n unordered marked points (Section 3). This means that: (a) the connection can be extended to the directions of variation of moduli, and (b) it is modular invariant.

This connection then gives rise to a monodromy morphism γ_{n} : Γ_{1,[n]} → G_{n} ⋊S_{n}, which we analyze in Section 4. The images of most generators can be expressed using the KZ associator, but the image Θ̃ of the S-transformation expresses using iterated integrals of Eisenstein series. The relations between generators give rise to relations between Θ̃ and the KZ associator, identities (28). This identity may be viewed as an elliptic analogue of the pentagon identity, as it is a “de Rham” analogue of the relation 6AS in [HLS00] (in [Man05], the question was asked of the existence of this kind of identity).

In Section 5, we investigate how to algebraically construct a morphism Γ_{1,[n]} → G_{n} ⋊ S_{n}. We show that a morphism \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n}) ⋊ S_{n} can be constructed using an associator only (here \overline{B}_{1,n} is the reduced braid group of n points on the torus). [Dri91] then implies that the formality isomorphism can be defined over ℚ. In the last part of Section 5, we develop the analogue of the theory of quasitriangular quasibialgebras (QTQBA’s), namely elliptic structures over QTQBA’s. These structures give rise to representations of \overline{B}_{1,n}, and they can be modified by twist. We hope that in the case of a simple Lie algebra, and using suitable twists, the elliptic structure given in Section 5.4 will give rise to elliptic structures over the quantum group U_{q}(𝔤) (where q ∈ ℂ^{×}) or over the Lusztig quantum group (when q is a root of unity), yielding back the representations of \overline{B}_{1,n} from conformal field theory.

In Section 6, we show that the universal KZB connection indeed specializes to the ordinary KZB connection.

Sections 7-9 are dedicated applications of the ideas of the preceding sections (in particular, Section 6) to representation theory of Cherednik algebras.

More precisely, In Section 7, we construct a homomorphism from the Lie algebra 𝔱̄_{1,n} ⋊ 𝔡 to the rational Cherednik algebra H_{n}(k) of type A_{n−1}. This allows us to consider the elliptic KZB connection with values in representations of the rational Cherednik algebra. The monodromy of this connection then gives representations of the true Cherednik algebra (i.e. the double affine Hecke algebra). In particular, this gives a simple way of constructing an isomorphism between the rational Cherednik algebra and the double affine Hecke algebra, with formal deformation parameters.

In Section 8, we consider the special representation V_{N} of the rational Cherednik algebra H_{n}(k), k = N/n, for which the elliptic KZB connection is the KZB connection for (holomorphic) n-point correlation functions of the WZW model for SL_{N}(ℂ) on the elliptic curve, when the marked points are labeled by the vector representation ℂ^{N}. This representation is realized in the space of equivariant polynomial functions on 𝔰𝔩_{N} with values in (ℂ^{N})^{⊗n}, and we show that it is irreducible, and calculate its character.

In Section 9, we generalize the construction of Section 8, by replacing, in the construction of V_{N}, the space of polynomial functions on 𝔰𝔩_{N} with an arbitrary D-module on 𝔰𝔩_{N}. This gives rise to an exact functor from the category of (equivariant) D-modules on 𝔰𝔩_{N} to the category of representations of H_{n}(N/n). We study this functor in detail. In particular, we show that this functor maps D-modules concentrated on the nilpotent cone to modules from category O_{−} of highest weight modules over the Cherednik algebra, and is closely related to the Gan-Ginzburg functor [GG04]. Using these facts, we show that it maps irreducible D-modules on the nilpotent cone to irreducible representations of the Cherednik algebra, and determine their highest weights. As an application, we compute the decomposition of cuspidal D-modules into irreducible representations of SL_{N}(ℂ). Finally, we describe the generalization of the above result to the trigonometric case (which involves D-modules on the group and trigonometric Cherednik algebras), and point out several directions for generalization.

1 Bundles with flat connections on (reduced) configuration spaces

1.1 The Lie algebras 𝔱_{1,n} and 𝔱̄_{1,n}

Let n ≥ 1 be an integer and k be a field of characteristic zero. We define 𝔱_{1,n}^{k} as the Lie algebra with generators x_{i}, y_{i} (i = 1, ..., n) and t_{ij} (i ≠ j ∈ {1, ..., n}) and relations

t_{ij} = t_{ji}, [t_{ij}, t_{ik} + t_{jk}] = 0, [t_{ij}, t_{kl}] = 0,  (1)

[x_{i}, y_{j}] = t_{ij}, [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0, [x_{i}, y_{i}] = − ∑_{j|j≠i} t_{ij},

[x_{i}, t_{jk}] = [y_{i}, t_{jk}] = 0, [x_{i} + x_{j}, t_{ij}] = [y_{i} + y_{j}, t_{ij}] = 0.

(i, j, k, l are distinct). In this Lie algebra, ∑_{i} x_{i} and ∑_{i} y_{i} are central; we then define 𝔱̄_{1,n}^{k} := 𝔱_{1,n}^{k}/(∑_{i} x_{i}, ∑_{i} y_{i}). Both 𝔱_{1,n}^{k} and 𝔱̄_{1,n}^{k} are positively graded, where deg(x_{i}) = deg(y_{i}) = 1.

The symmetric group S_{n} acts by automorphisms of 𝔱_{1,n}^{k} by σ(x_{i}) := x_{σ(i)}, σ(y_{i}) := y_{σ(i)}, σ(t_{ij}) := t_{σ(i)σ(j)}; this induces an action of S_{n} by automorphisms of 𝔱̄_{1,n}^{k}.

We will set 𝔱_{1,n} := 𝔱_{1,n}^{ℂ}, 𝔱̄_{1,n} := 𝔱̄_{1,n}^{ℂ} in Sections 1 to 4.

1.2 Bundles with flat connections over C(E, n) and C̄(E, n)

Let E be an elliptic curve, C(E, n) be the configuration space E^{n}−{diagonals} (n ≥ 1) and C̄(E, n) := C(E, n)/E be the reduced configuration space. We will define a ^{4} exp(ˆ𝔱̄_{1,n})-principal bundle with a flat (holomorphic) connection (P̄_{E,n}, ¯∇_{E,n}) → C̄(E, n). For this, we define a exp(𝔱̂_{1,n})-principal bundle with a flat connection (P_{E,n}, ∇_{E,n}) → C(E, n). Its image under the natural morphism exp(𝔱̂_{n}) → exp(ˆ𝔱̄_{n}) is a exp(ˆ𝔱̄_{1,n})-bundle with connection (P̃_{E,n}, ˜∇_{E,n}) → C(E, n), and we then prove that (P̃_{E,n}, ˜∇_{E,n}) is the pull-back of a pair (P̄_{E,n}, ¯∇_{E,n}) under the canonical projection C(E, n) → C̄(E, n).

For this, we fix a uniformization E ≃ E_{τ}, where for τ ∈ ℌ, ℌ := {τ ∈ ℂ|ℑ(τ) > 0}, E_{τ} := ℂ/Λ_{τ} and Λ_{τ} := ℤ + ℤτ. We then have C(E_{τ} , n) = (ℂ^{n} − Diag_{n,τ})/Λ_{τ}^{n}, where

Diag_{n,τ} := {z = (z_{1}, ..., z_{n}) ∈ ℂ^{n} | z_{ij} := z_{i} − z_{j} ∈ Λ_{τ} for some i ≠ j}.

We define P_{τ,n} as the restriction to C(E_{τ} , n) of the bundle over ℂ^{n}/Λ_{τ}^{n} for which a section on U ⊂ ℂ^{n}/Λ_{τ}^{n} is a regular map f : π^{−1}(U) → exp(𝔱̂_{1,n}), such that^{5} f(z + δ_{i}) = f(z), f(z + τδ_{i}) = e^{−2πix_{i}}f(z) (here π : ℂ^{n} → ℂ^{n}/Λ_{τ}^{n} is the canonical projection and δ_{i} is the ith vector of the canonical basis of ℂ^{n}).

The bundle P̃_{τ,n} → C(E_{τ} , n) derived from P_{τ,n} is the pull-back of a bundle P̄_{τ,n} → C̄(E_{τ} , n) since the e^{−2πix̄_{i}} ∈ exp(ˆ𝔱̄_{1,n}) commute pairwise and their product is 1. Here x ↦ x̄ is the map 𝔱̂_{1,n} → ˆ 𝔱̄_{1,n}.

A flat connection ∇_{τ,n} on P_{τ,n} is then the same as an equivariant flat connection over the trivial bundle over ℂ^{n} − Diag_{n,τ}, i.e., a connection of the form

∇_{τ,n} := d − ∑_{i=1}^{n} K_{i}(z|τ)dz_{i},

where K_{i}(−|τ) : ℂ^{n} → 𝔱̂_{1,n} is holomorphic on ℂ^{n} − Diag_{n,τ}, such that: (a) K_{i}(z + δ_{j}|τ) = K_{i}(z|τ), K_{i}(z + τδ_{j}|τ) = e^{−2πiad(x_{j})}(K_{i}(z|τ)),

(b) [∂/∂z_{i} − K_{i}(z|τ), ∂/∂z_{j} − K_{j}(z|τ)] = 0 for any i, j.

4We will denote by 𝔤̂ or 𝔤^{∧} the degree completion of a positively graded Lie algebra 𝔤.

5We set i := \sqrt{−1}, leaving i for indices.

∇_{τ,n} then induces a flat connection ˜∇_{τ,n} on P̃_{τ,n}. Then ˜∇_{τ,n} is the pullback of a (necessarily flat) connection on P̄_{τ,n} iff:

(c) K̄_{i}(z|τ) = K̄_{i}(z+u(∑_{i} δ_{i})|τ) and ∑_{i} K̄_{i}(z|τ) = 0 for z ∈ ℂ^{n} −Diag_{n,τ}, u ∈ ℂ.

In order to define the K_{i}(z|τ), we first recall some facts on theta-functions. There is a unique holomorphic function ℂ×ℌ → ℂ, (z, τ) ↦ θ(z|τ), such that

• {z|θ(z|τ) = 0} = Λ_{τ},

• θ(z + 1|τ) = −θ(z|τ) = θ(−z|τ),

• θ(z + τ|τ) = −e^{−πiτ} e^{−2πiz}θ(z|τ), and

• θ_{z}(0|τ) = 1.

We have θ(z|τ + 1) = θ(z|τ), while θ(−z/τ| − 1/τ) = −(1/τ)e^{(πi/τ)z^{2}}θ(z|τ). If η(τ) = q^{1/24} ∏_{n≥1}(1−q^{n}) where q = e^{2πiτ}, and if we set ϑ(z|τ) := η(τ)^{3}θ(z|τ), then ∂_{τ} ϑ = (1/4πi)∂_{z}^{2}ϑ.

Let us set

k(z, x|τ) := \frac{θ(z + x|τ)}{θ(z|τ)θ(x|τ)} − \frac{1}{x}.

When τ is fixed, k(z, x|τ) belongs to Hol(ℂ − Λ_{τ})[[x]]. Substituting x = adx_{i}, we get a linear map 𝔱_{1,n} → (𝔱_{1,n} ⊗ Hol(ℂ − Λ_{τ}))^{∧}, and taking the image of t_{ij}, we define

K_{ij}(z|τ) := k(z, adx_{i}|τ)(t_{ij}) = (\frac{θ(z + ad(x_{i})|τ)}{θ(z|τ)} \frac{ad(x_{i})}{θ(ad(x_{i})|τ)} − 1)(y_{j});

it is a holomorphic function on ℂ − Λ_{τ} with values in 𝔱̂_{1,n}.

Now set z := (z_{1}, …, z_{n}), z_{ij} := z_{i} − z_{j} and define

K_{i}(z|τ) := −y_{i} + ∑_{j|j≠i} K_{ij}(z_{ij}|τ).

Let us check that the K_{i}(z|τ) satisfy condition (c). We have clearly K_{i}(z+u(∑_{i} δ_{i})) = K_{i}(z). We have k(z, x|τ)+k(−z, −x|τ) = 0, so K_{ij}(z|τ)+ K_{ji}(−z|τ) = 0, so that ∑_{i} K_{i}(z|τ) = − ∑_{i} y_{i}, which implies ∑_{i} K̄_{i}(z|τ) = 0.

Lemma 1. K_{i}(z + δ_{j}|τ) = K_{i}(z|τ) and K_{i}(z + τδ_{j}|τ) = e^{−2πiadx_{j}}(K_{i}(z|τ)), i.e., the K_{i}(z|τ) satisfy condition (a).

Proof. We have k(z ± 1, x|τ) = k(z, x|τ) so for any j, K_{i}(z + δ_{j}|τ) = K_{i}(z|τ). We have k(z ± τ, x|τ) = e^{∓2πix}k(z, x|τ) + (e^{∓2πix} − 1)/x, so if j ≠ i,

K_{i}(z + τδ_{j}|τ) = ∑_{j′≠i,j} K_{ij′}(z_{ij′}|τ) + e^{2πiadx_{i}}K_{ij}(z_{ij}|τ) + \frac{e^{2πiadx_{i}} − 1}{adx_{i}}(t_{ij}) − y_{i}.

Then

\frac{e^{2πiadx_{i}} − 1}{adx_{i}}(t_{ij}) = \frac{1 − e^{−2πiadx_{j}}}{adx_{j}} (t_{ij}) = (1 − e^{−2πiadx_{j}})(y_{i}),

e^{2πiadx_{i}}(K_{ij}(z_{ij}|τ)) = e^{−2πiadx_{j}}(K_{ij}(z_{ij}|τ)) and for j′ ≠ i, j, K_{ij′}(z_{ij′}|τ) = e^{−2πiadx_{j}}(K_{ij′}(z_{ij′}|τ)), so K_{i}(z + τδ_{j}|τ) = e^{−2πiadx_{j}}(K_{i}(z|τ)). Now

K_{i}(z + τδ_{i}|τ) = − ∑_{i} y_{i} − ∑_{j|j≠i} K_{j}(z + τδ_{i}|τ)
= − ∑_{i} y_{i} − e^{−2πiadx_{i}}( ∑_{j|j≠i} K_{j}(z|τ))
= e^{−2πiadx_{i}}(− ∑_{i} y_{i} − ∑_{j|j≠i} K_{j}(z|τ))
= e^{−2πiadx_{i}}K_{i}(z|τ)

(the first and last equality follow from the proof of (c), the second equality has just been proved, the third equality follows from the centrality of ∑_{i} y_{i}). ⊓ ⊔

Proposition 2. [∂/∂z_{i} − K_{i}(z|τ), ∂/∂z_{j} − K_{j}(z|τ)] = 0, i.e., the K_{i}(z|τ) satisfy condition (b).

Proof. For i ≠ j, let us set K_{ij} := K_{ij}(z_{ij}|τ). Recall that K_{ij} + K_{ji} = 0, therefore if ∂_{i} := ∂/∂z_{i}

∂_{i}K_{ij} − ∂_{j}K_{ji} = 0, [y_{i} − K_{ij}, y_{j} − K_{ji}] = −[K_{ij}, y_{i} + y_{j}].

Moreover, if i, j, k, l are distinct, then [K_{ik}, K_{jl}] = 0. It follows that if i ≠ j, then [∂_{i} − K_{i}(z|τ), ∂_{j} − K_{j}(z|τ)] equals

[y_{i}+y_{j}, K_{ij}]+ ∑_{k|k≠i,j} ([K_{ik}, K_{jk}]+[K_{ij}, K_{jk}]+[K_{ij}, K_{ik}]+[y_{j}, K_{ik}]−[y_{i}, K_{jk}]).

Let us assume for a while that if k ∉ {i, j}, then

−[y_{i}, K_{jk}] − [y_{j}, K_{ki}] − [y_{k}, K_{ij}] + [K_{ji}, K_{ki}] + [K_{kj}, K_{ij}] + [K_{ik}, K_{jk}] = 0 (2)

(this is the universal version of the classical dynamical Yang-Baxter equation). Then (2) implies that

[∂_{i} −K_{i}(z|τ), ∂_{j} −K_{j}(z|τ)] = [y_{i} +y_{j}, K_{ij}]+ ∑_{k|k≠i,j}[y_{k}, K_{ij}] = [∑_{k} y_{k}, K_{ij}] = 0

(as ∑_{k} y_{k} is central), which proves the proposition.

Let us now prove (2). If f(x) ∈ ℂ[[x]], then

[y_{k}, f(adx_{i})(t_{ij})] = \frac{f(adx_{i}) − f(−adx_{j})}{adx_{i} + adx_{j}}[−t_{ki}, t_{ij}],

[y_{i}, f(adx_{j})(t_{jk})] = \frac{f(adx_{j}) − f(−adx_{k})}{adx_{j} + adx_{k}}[−t_{ij}, t_{jk}]
= \frac{f(adx_{j}) − f(adx_{i} + adx_{j})}{−adx_{i}}[−t_{ij}, t_{jk}],

[y_{j}, f(adx_{k})(t_{ki})] = \frac{f(adx_{k}) − f(−adx_{i})}{adx_{k} + adx_{i}}[−t_{jk}, t_{ki}]
= \frac{f(−adx_{i} − adx_{j}) − f(−adx_{i})}{−adx_{j}}[−t_{jk}, t_{ki}].

The first identity is proved as follows:

[y_{k}, (adx_{i})^{n}(t_{ij})] = − ∑_{s=0}^{n−1}(adx_{i})^{s}(adt_{ki})(adx_{i})^{n−1−s}(t_{ij})

= − ∑_{s=0}^{n−1}(adx_{i})^{s}(adt_{ki})(−adx_{j})^{n−1−s}(t_{ij})

= − ∑_{s=0}^{n−1}(adx_{i})^{s}(−adx_{j})^{n−1−s}(adt_{ki})(t_{ij})
= f(adx_{i}, −adx_{j})([−t_{ki}, t_{ij}]),

where f(u, v) = (u^{n} − v^{n})/(u − v). The two next identities follow from this one and from the fact that x_{i} + x_{j} + x_{k} commutes with t_{ij}, t_{ik}, t_{jk}.

Then, if we write k(z, x) instead of k(z, x|τ), the l.h.s. of (2) is equal to

(k(z_{ij}, −adx_{j})k(z_{ik}, adx_{i} + adx_{j}) − k(z_{ij}, adx_{i})k(z_{jk}, adx_{i} + adx_{j})

+k(z_{ik}, adx_{i})k(z_{jk}, adx_{j}) + \frac{k(z_{jk}, adx_{j}) − k(z_{jk}, adx_{i} + adx_{j})}{adx_{i}}

+\frac{k(z_{ik}, adx_{i}) − k(z_{ij}, adx_{i} + adx_{j})}{adx_{j}} − \frac{k(z_{ij}, adx_{i}) − k(z_{ij}, −adx_{j})}{adx_{i} + adx_{j}})[t_{ij}, t_{ik}].

So (2) follows from the identity

k(z, −v)k(z′, u + v) − k(z, u)k(z′ − z, u + v) + k(z′, u)k(z′ − z, v)

+\frac{k(z′ − z, v) − k(z′ − z, u + v)}{u}+\frac{k(z′, u) − k(z′, u + v)}{v}−\frac{k(z, u) − k(z, −v)}{u + v} = 0,

where u, v are formal variables, which is a consequence of the theta-functions identity

(k(z, −v) − \frac{1}{v} )(k(z′, u + v) + \frac{1}{u + v} ) − (k(z, u) + \frac{1}{u})(k(z′ − z, u + v) + \frac{1}{u + v} )

+(k(z′, u) + \frac{1}{u})(k(z′ − z, v) + \frac{1}{v} ) = 0.  (3)
⊓ ⊔

We have therefore proved:

Theorem 3. (P_{τ,n}, ∇_{τ,n}) is a flat connection on C(E_{τ} , n), and the induced flat connection (P̃_{τ,n}, ˜∇_{τ,n}) is the pull-back of a unique flat connection (P̄_{τ,n}, ¯∇_{τ,n}) on C̄(E_{τ} , n).

1.3 Bundles with flat connections on C(E, n)/S_{n} and C̄(E, n)/S_{n}

The group S_{n} acts freely by automorphisms of C(E, n) by σ(z_{1}, ..., z_{n}) := (z_{σ^{−1}(1)}, ..., z_{σ^{−1}(n)}). This descends to a free action of S_{n} on C̄(E, n). We set C(E, [n]) := C(E, n)/S_{n}, C̄(E, [n]) := C̄(E, n)/S_{n}.

We will show that (P_{τ,n}, ∇_{τ,n}) induces a bundle with flat connection (P_{τ,[n]}, ∇_{τ,[n]}) on C(E_{τ} , [n]) with group exp(𝔱̂_{1,n})⋊S_{n}, and similarly (P̄_{τ,n}, ¯∇_{τ,n}) induces (P̄_{τ,[n]}, ¯∇_{τ,[n]}) on C̄(E_{τ} , [n]) with group exp(ˆ𝔱̄_{1,n}) ⋊ S_{n}.

We define P_{τ,[n]} → C(E_{τ} , [n]) by the condition that a section of U ⊂ C(E_{τ} , [n]) is a regular map π^{−1}(U) → exp(𝔱̂_{1,n}) ⋊ S_{n}, satisfying again f(z + δ_{i}) = f(z), f(z + τδ_{i}) = e^{−2πix_{i}}f(z) and the additional requirement f(σz) = σf(z) (where π̃ : ℂ^{n} − Diag_{τ,n} → C(E_{τ} , [n]) is the canonical projection). It is clear that ∇_{τ,n} is S_{n}-invariant, which implies that it defines a flat connection ∇_{τ,[n]} on C(E_{τ} , [n]).

The bundle P̄(E_{τ} , [n]) → C̄(E_{τ} , [n]) is defined by the additional requirement f(z + u(∑_{i} δ_{i})) = f(z) and ¯∇_{τ,n} then induces a flat connection ¯∇_{τ,[n]} on C̄(E_{τ} , [n]).

2 Formality of pure braid groups on the torus

2.1 Reminders on Malcev Lie algebras

Let k be a field of characteristic 0 and let 𝔤 be a pronilpotent k-Lie algebra. Set 𝔤^{1} = 𝔤, 𝔤^{k+1} = [𝔤, 𝔤^{k}]; then 𝔤 = 𝔤^{1} ⊃ 𝔤^{2}... is a decreasing filtration of 𝔤. The associated graded Lie algebra is gr(𝔤) := ⊕_{k≥1}𝔤^{k}/𝔤^{k+1}; we also consider its completion ĝr(𝔤) := ˆ⊕_{k≥1}𝔤^{k}/𝔤^{k+1} (here ˆ⊕ is the direct product). We say that 𝔤 is formal iff there exists an isomorphism of filtered Lie algebras 𝔤 ≃ ĝr(𝔤), whose associated graded morphism is the identity. We will use the following fact: if 𝔤 is a pronilpotent Lie algebra, 𝔱 is a positively graded Lie algebra, and there exists an isomorphism 𝔤 ≃ 𝔱̂ of filtered Lie algebras, then 𝔤 is formal, and the associated graded morphism gr(𝔤) → 𝔱 is an isomorphism of graded Lie algebras.

If Γ is a finitely generated group, there exists a unique pair (Γ (k), i_{Γ}) of a prounipotent algebraic group Γ (k) and a group morphism i_{Γ} : Γ → Γ (k), which is initial in the category of all pairs (U, j), where U is prounipotent k-algebraic group and j : Γ → U is a group morphism.

We denote by Lie(Γ)_{k} the Lie algebra of Γ (k). Then we have Γ (k) = exp(Lie(Γ)_{k}); Lie(Γ)_{k} is a pronilpotent Lie algebra. We have Lie(Γ)_{k} = Lie(Γ)_{ℚ} ⊗ k. We say that Γ is formal iff Lie(Γ)_{ℂ} is formal (one can show that this implies that Lie(Γ)_{ℚ} is formal).

When Γ is presented by generators g_{1}, ..., g_{n} and relations R_{i}(g_{1}, ..., g_{n}) (i = 1, ..., p), Lie(Γ)_{ℚ} is the quotient of the topologically free Lie algebra 𝔣̂_{n} generated by γ_{1}, ..., γ_{n} by the topological ideal generated by log(R_{i}(e^{γ_{1}}, ..., e^{γ_{n}})) (i = 1, ..., p).

The decreasing filtration of 𝔣̂_{n} is 𝔣̂_{n} = (𝔣̂_{n})^{1} ⊃ (𝔣̂_{n})^{2} ⊃ ..., where (𝔣̂_{n})^{k} is the part of 𝔣̂_{n} of degree ≥ k in the generators γ_{1}, ..., γ_{n}. The image of this filtration by the projection is map is the decreasing filtration Lie(Γ)_{ℚ} = Lie(Γ)_{ℚ}^{1} ⊃ Lie(Γ)_{ℚ}^{2} ⊃ ... of Lie(Γ)_{ℚ}.

2.2 Presentation of PB_{1,n}

For τ ∈ ℌ, let U_{τ} ⊂ ℂ^{n} − Diag_{n,τ} be the open subset of all z = (z_{1}, ..., z_{n}), of the form z_{i} = a_{i} + τb_{i}, where 0 < a_{1} < ... < a_{n} < 1 and 0 < b_{1} < ... < b_{n} < 1. If z_{0} = (z_{1}^{0}, ..., z_{n}^{0}) ∈ U_{τ}, its image \overline{z}_{0} in E_{τ}^{n} actually belongs to the configuration space C(E_{τ} , n).

The pure braid group of n points on the torus PB_{1,n} may be viewed as PB_{1,n} = π_{1}(C(E_{τ} , n), \overline{z}_{0}). Denote by X_{i}, Y_{i} ∈ PB_{1,n} the classes of the projection of the paths [0, 1] ∋ t ↦ z_{0} − tδ_{i} and [0, 1] ∋ t ↦ z_{0} − tτδ_{i}.

Set A_{i} := X_{i}...X_{n}, B_{i} := Y_{i}...Y_{n} for i = 1, ..., n. According to [Bir69a], A_{i}, B_{i} (i = 1, ..., n) generate PB_{1,n} and a presentation of PB_{1,n} is, in terms of these generators:

(A_{i}, A_{j}) = (B_{i}, B_{j}) = 1 (any i, j), (A_{1}, B_{j}) = (B_{1}, A_{j}) = 1 (any j),
(B_{k}, A_{k}A_{j}^{−1}) = (B_{k}B_{j}^{−1}, A_{k}) = C_{jk} (j ≤ k);
(A_{i}, C_{jk}) = (B_{i}, C_{jk}) = 1 (i ≤ j ≤ k),

where (g, h) = ghg^{−1}h^{−1}.

2.3 Alternative presentations of 𝔱_{1,n}

We now give two variants of the defining presentation of 𝔱_{1,n}. Presentation (A) below is the original presentation in [Bez94], and presentation (B) will be suited to the comparison with the above presentation of PB_{1,n}.

Lemma 4. 𝔱_{1,n} admits the following presentations:

(A) generators are x_{i}, y_{i} (i = 1, ..., n), relations are [x_{i}, y_{j}] = [x_{j}, y_{i}] (i ≠ j), [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0 (any i, j), [∑_{j} x_{j}, y_{i}] = [∑_{j} y_{j}, x_{i}] = 0 (any i), [x_{i}, [x_{j}, y_{k}]] = [y_{i}, [y_{j}, x_{k}]] = 0 (i, j, k are distinct);

(B) generators are a_{i}, b_{i} (i = 1, ..., n), relations are [a_{i}, a_{j}] = [b_{i}, b_{j}] = 0 (any i, j), [a_{1}, b_{j}] = [b_{1}, a_{j}] = 0 (any j), [a_{j}, b_{k}] = [a_{k}, b_{j}] (any i, j), [a_{i}, c_{jk}] = [b_{i}, c_{jk}] = 0 (i ≤ j ≤ k), where c_{jk} = [b_{k}, a_{k} − a_{j}].

The isomorphism of presentations (A) and (B) is a_{i} = ∑_{j=i}^{n} x_{j}, b_{i} = ∑_{j=i}^{n} y_{j}.

Proof. Let us prove that the initial relations for x_{i}, y_{i}, t_{ij} imply the relations (A) for x_{i}, y_{i}. Let us assume the initial relations. If i ≠ j, since [x_{i}, y_{j}] = t_{ij} and t_{ij} = t_{ji}, we get [x_{i}, y_{j}] = [x_{j}, y_{i}]. The relations [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0 (any i, j) are contained in the initial relations. For any i, since [x_{i}, y_{i}] = − ∑_{j|j≠i} t_{ij} and [x_{j}, y_{i}] = t_{ji} = t_{ij} (j ≠ i), we get [∑_{j} x_{j}, y_{i}] = 0. Similarly, [∑_{j} y_{j}, x_{i}] = 0 (for any i). If i, j, k are distinct, since [x_{j}, y_{k}] = t_{jk} and [x_{i}, t_{jk}] = 0, we get [x_{i}, [x_{j}, y_{k}]] = 0 and similarly we prove [x_{i}, [y_{j}, x_{k}]] = 0.

Let us now prove that the relations (A) for x_{i}, y_{i} imply the initial relations for x_{i}, y_{i} and t_{ij} := [x_{i}, y_{j}] (i ≠ j). Assume the relations (A). If i ≠ j, since [x_{i}, y_{j}] = [x_{j}, y_{i}], we have t_{ij} = t_{ji}. The relation t_{ij} = [x_{i}, y_{j}] (i ≠ j) is clear and [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0 (any i, j) are already in relations (A). Since for any i, [∑_{j} x_{j}, y_{i}] = 0, we get [x_{i}, y_{i}] = − ∑_{j|j≠i}[x_{j}, y_{i}] = − ∑_{j|j≠i} t_{ji} = − ∑_{j|j≠i} t_{ij}. If i, j, k are distinct, the relations [x_{i}, [x_{j}, y_{k}]] = [y_{i}, [y_{j}, x_{k}]] = 0 imply [x_{i}, t_{jk}] = [y_{i}, t_{jk}] = 0. If i ≠ j, since [∑_{k} x_{k}, x_{i}] = [∑_{k} x_{k}, y_{j}] = 0, we get [∑_{k} x_{k}, t_{ij}] = 0 and [x_{k}, t_{ij}] = 0 for k ∉ {i, j} then implies [x_{i} + x_{j}, t_{ij}] = 0. One proves similarly [y_{i} + y_{j}, t_{ij}] = 0. We have already shown that [x_{i}, t_{kl}] = [y_{j}, t_{kl}] = 0 for i, j, k, l distinct, which implies [[x_{i}, y_{j}], t_{kl}] = 0, i.e., [t_{ij}, t_{kl}] = 0. If i, j, k are distinct, we have shown that [t_{ij}, y_{k}] = 0 and [t_{ij}, x_{i} + x_{j}] = 0, which implies [t_{ij}, [x_{i} + x_{j}, y_{k}]] = 0, i.e., [t_{ij}, t_{ik} + t_{jk}] = 0.

Let us prove that the relations (A) for x_{i}, y_{i} imply relations (B) for a_{i} := ∑_{j=i}^{n} x_{j}, b_{i} := ∑_{j=i}^{n} y_{j}. Summing up the relations [x_{i′}, x_{j′}] = [y_{i′}, y_{j′}] = 0 and [x_{i′}, y_{j′}] = [x_{j′}, y_{i′}] for i′ = i, ..., n and j′ = j, ..., n, we get [a_{i}, a_{j}] = [b_{i}, b_{j}] = 0 and [a_{i}, b_{j}] = [a_{j}, b_{i}] (for any i, j). Summing up [∑_{j} x_{j}, y_{i′}] = [∑_{j} y_{j}, x_{i′}] = 0 for i′ = i, ..., n, we get [a_{1}, b_{i}] = [a_{i}, b_{1}] = 0 (for any i). Finally, c_{jk} = ∑_{α=j}^{k−1} ∑_{β=k}^{n} t_{αβ} (in terms of the initial presentation) so the relations [x_{i′}, t_{αβ}] = 0 for i′ ≠ α, β and [x_{α} + x_{β}, t_{αβ}] = 0 imply [a_{i}, c_{jk}] = 0 for i ≤ j ≤ k. Similarly, one shows [b_{i}, c_{jk}] = 0 for i ≤ j ≤ k.

Let us prove that the relations (B) for a_{i}, b_{i} imply relations (A) for x_{i} := a_{i} − a_{i+1}, y_{i} := b_{i} − b_{i+1} (with the convention a_{n+1} = b_{n+1} = 0). As before, [a_{i}, a_{j}] = [b_{i}, b_{j}] = 0, [a_{i}, b_{j}] = [a_{j}, b_{i}] imply [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0, [x_{i}, y_{j}] = [x_{j}, y_{i}] (for any i, j). We set t_{ij} := [x_{i}, y_{j}] for i ≠ j, then we have t_{ij} = t_{ji}. We have for j < k, t_{jk} = c_{jk} − c_{j,k+1} − c_{j+1,k} + c_{j+1,k+1} (we set c_{i,n+1} := 0), so [a_{i}, c_{jk}] = 0 implies [∑_{i′=i}^{n} x_{i′}, t_{jk}] = 0 for i ≤ j < k. When i < j < k, the difference between this relation and its analogue of (i+1, j, k) gives [x_{i}, t_{jk}] = 0 for i < j < k. This can be rewritten [x_{i}, [x_{j}, y_{k}]] = 0 and since [x_{i}, x_{j}] = 0, we get [x_{j}, [x_{i}, y_{k}]] = 0, so [x_{j}, t_{ik}] = 0 and by changing indices, [x_{i}, t_{jk}] = 0 for j < i < k. Rewriting again [x_{i}, t_{jk}] = 0 for i < j < k as [x_{i}, [y_{j}, x_{k}]] = 0 and using [x_{i}, x_{k}] = 0, we get [x_{k}, [x_{i}, y_{j}]] = 0. i.e., [x_{k}, t_{ij}] = 0, which we rewrite [x_{i}, t_{jk}] = 0 for j < k < i. Finally, [x_{i}, t_{jk}] = 0 for j < k and i ∉ {j, k}, which implies [x_{i}, t_{jk}] = 0 for i, j, k different. One proves similarly [y_{i}, t_{jk}] = 0 for i, j, k different. ⊓ ⊔

2.4 The formality of PB_{1,n}

The flat connection d − ∑_{i=1}^{n} K_{i}(z|τ)dz_{i} gives rise to a monodromy representation

μ_{z_{0},τ} : PB_{1,n} = π_{1}(C, \overline{z}_{0}) → exp(𝔱̂_{1,n}),

which factors through a morphism μ_{z_{0},τ} (ℂ) : PB_{1,n}(ℂ) → exp(𝔱̂_{1,n}). Let Lie(μ_{z_{0},τ}) : Lie(PB_{1,n})_{ℂ} → 𝔱̂_{1,n} be the corresponding morphism between pronilpotent Lie algebras.

Proposition 5. Lie(μ_{z_{0},τ}) is an isomorphism of filtered Lie algebras, so that PB_{1,n} is formal.

Proof. As we have seen, Lie(PB_{1,n})_{ℂ} (denoted Lie(PB_{1,n}) in this proof) is the quotient of the topologically free Lie algebra generated by α_{i}, β_{i} (i = 1, ..., n) by the topological ideal generated by [α_{i}, α_{j}], [β_{i}, β_{j}], [α_{1}, β_{j}], [β_{1}, α_{j}], log(e^{β_{k}}, e^{α_{k}−α_{j}}) − log(e^{β_{k}−β_{j}}, e^{α_{k}}), [α_{i}, γ_{jk}], [β_{i}, γ_{jk}] where γ_{jk} = log(e^{β_{k}}, e^{α_{k}−α_{j}}).

This presentation and the above presentation (B) of 𝔱_{1,n} imply that there is a morphism of graded Lie algebras p_{n} : 𝔱_{1,n} → grLie(PB_{1,n}) defined by a_{i} ↦ [α_{i}], b_{i} ↦ [β_{i}], where α ↦ [α] is the projection map Lie(PB_{1,n}) → gr_{1}Lie(PB_{1,n}).

p_{n} is surjective because grLieΓ is generated in degree 1 (as the associated graded of any quotient of a topologically free Lie algebra).

There is a unique derivation Δ̃_{0} ∈ Der(𝔱_{1,n}), such that Δ̃_{0}(x_{i}) = y_{i} and Δ̃_{0}(y_{i}) = 0. This derivation gives rise to a one-parameter group of automorphisms of Der(𝔱_{1,n}), defined by exp(sΔ̃_{0})(x_{i}) := x_{i} + sy_{i}, exp(sΔ̃_{0})(y_{i}) = y_{i}.

Lie(μ_{z_{0},τ}) induces a morphism grLie(μ_{z_{0},τ}) : grLie(PB_{1,n}) → 𝔱_{1,n}. We will now prove that

grLie(μ_{z_{0},τ}) ∘ p_{n} = exp(−\frac{τ}{2πi}Δ̃_{0}) ∘ w,  (4)

where w is the automorphism of 𝔱_{1,n} defined by w(a_{i}) = −b_{i}, w(b_{i}) = 2πia_{i}. μ_{z_{0},τ} is defined as follows. Let F_{z_{0}}(z) be the solution of

(∂/∂z_{i})F_{z_{0}}(z) = K_{i}(z|τ)F_{z_{0}}(z), F_{z_{0}}(z_{0}) = 1

on U_{τ}; let

H_{τ} := {z = (z_{1}, ..., z_{n}) | z_{i} = a_{i} + τb_{i}, 0 < a_{1} < ... < a_{n} < 1}

and

V_{τ} := {z = (z_{1}, ..., z_{n}) | z_{i} = a_{i} + τb_{i}, 0 < b_{1} < ... < b_{n} < 1};

let F_{z_{0}}^{H} and F_{z_{0}}^{V} be the analytic prolongations of F_{z_{0}} to H_{τ} and V_{τ}; then

F_{z_{0}}^{H}(z + δ_{i}) = F_{z_{0}}^{H}(z)μ_{z_{0},τ} (X_{i}), e^{2πix_{i}}F_{z_{0}}^{V}(z + τδ_{i}) = F_{z_{0}}^{V}(z)μ_{z_{0},τ} (Y_{i}).

We have logF_{z_{0}}(z) = − ∑_{i}(z_{i} − z_{i}^{0})y_{i} + terms of degree ≥ 2, where 𝔱_{1,n} is graded by deg(x_{i}) = deg(y_{i}) = 1, which implies that logμ_{z_{0},τ} (X_{i}) = −y_{i} + terms of degree ≥ 2, logμ_{z_{0},τ} (Y_{i}) = 2πix_{i} − τy_{i} + terms of degree ≥ 2. Therefore Lie(μ_{z_{0},τ})(α_{i}) = logμ_{z_{0},τ} (A_{i}) = −b_{i} + terms of degree ≥ 2, Lie(μ_{z_{0},τ})(β_{i}) = logμ_{z_{0},τ} (B_{i}) = 2πia_{i} − τb_{i} + terms of degree ≥ 2. So grLie(μ_{z_{0},τ})([α_{i}]) = −b_{i}, grLie(μ_{z_{0},τ})([β_{i}]) = 2πia_{i} − τb_{i}.

It follows that grLie(μ_{z_{0},τ}) ∘ p_{n} is the endomorphism a_{i} ↦ −b_{i}, b_{i} ↦ 2πia_{i} − τb_{i} of 𝔱_{1,n}, which is the automorphism exp(−\frac{τ}{2πi} Δ̃_{0}) ∘ w; this proves (4).

Since we already proved that p_{n} is surjective, it follows that grLie(μ_{z_{0},τ}) and p_{n} are both isomorphisms. As Lie(PB_{1,n}) and 𝔱̂_{1,n} are both complete and separated, Lie(μ_{z_{0},τ}) is bijective, and since it is a morphism, it is an isomorphism of filtered Lie algebras. ⊓ ⊔

2.5 The formality of \overline{PB}_{1,n}

Let z_{0} ∈ U_{τ} and [z_{0}] ∈ C̄(E_{τ} , n) be its image. We set

\overline{PB}_{1,n} := π_{1}(C̄(E_{τ} , n), [z_{0}]).

Then \overline{PB}_{1,n} is the quotient of PB_{1,n} by its central subgroup (isomorphic to ℤ^{2}) generated by A_{1} and B_{1}. We have μ_{z_{0},τ} (A_{1}) = e^{− ∑_{i} y_{i}} and μ_{z_{0},τ} (B_{1}) = e^{2πi ∑_{i} x_{i}−τ ∑_{i} y_{i}}, so Lie(μ_{z_{0},τ})(α_{1}) = −a_{1}, Lie(μ_{z_{0},τ})(β_{1}) = 2πia_{1} −τb_{1}, which implies that Lie(μ_{z_{0},τ}) induces an isomorphism between Lie(\overline{PB}_{1,n})_{ℂ} and 𝔱̄_{1,n}. In particular, \overline{PB}_{1,n} is formal.

Remark 6. Let Diag_{n} := {(z, τ) ∈ ℂ^{n} × ℌ|z ∈ Diag_{n,τ} } and let U ⊂ (ℂ^{n} × ℌ) − Diag_{n} be the set of all (z, τ) such that z ∈ U_{τ}. Each element of U gives rise to a Lie algebra isomorphism μ_{z,τ} : Lie(PB_{1,n}) ≃ 𝔱̂_{1,n}. For an infinitesimal (dz, dτ), the composition μ_{z+dz,τ+dτ} ∘ μ_{z,τ}^{−1} is then an infinitesimal automorphism of 𝔱̂_{1,n}. This defines a flat connection over U with values in the trivial Lie algebra bundle with Lie algebra Der(𝔱̂_{1,n}). When dτ = 0, the infinitesimal automorphism has the form exp(∑_{i} K_{i}(z|τ)dz_{i}), so the connection has the form d − ∑_{i} ad(K_{i}(z|τ))dz_{i} − Δ̃(z|τ)dτ, where Δ̃ : U → Der(𝔱̂_{1,n}) is a meromorphic map with poles at Diag_{n}. In the next section, we determine a map Δ : (ℂ^{n} × ℌ) − Diag_{n} → Der(𝔱̂_{1,n}) with the same flatness properties as Δ̃(z|τ).

2.6 The isomorphisms B_{1,n}(ℂ) ≃ exp(𝔱̂_{1,n}) ⋊ S_{n}, \overline{B}_{1,n}(ℂ) ≃ exp(ˆ𝔱̄_{1,n}) ⋊ S_{n}

Let z_{0} be as above; we define B_{1,n} := π_{1}(C(E_{τ} , [n]), [z_{0}]) and \overline{B}_{1,n} := π_{1}(C̄(E_{τ} , [n]), [\overline{z}_{0}]), where x ↦ [x] is the canonical projection C(E_{τ} , n) → C(E_{τ} , [n]) or C̄(E_{τ} , n) → C̄(E_{τ} , [n]).

We have an exact sequence 1 → PB_{1,n} → B_{1,n} → S_{n} → 1, We then define groups B_{1,n}(ℂ) fitting in an exact sequence 1 → PB_{1,n}(ℂ) → B_{1,n}(ℂ) → S_{n} → 1 as follows: the morphism B_{1,n} → Aut(PB_{1,n}) extends to B_{1,n} → Aut(PB_{1,n}(ℂ)); we then construct the semidirect product PB_{1,n}(ℂ) ⋊ B_{1,n}; then PB_{1,n} embeds diagonally as a normal subgroup of this semidirect product, and B_{1,n}(ℂ) is defined as the quotient (PB_{1,n}(ℂ) ⋊ B_{1,n})/PB_{1,n}.

The monodromy of ∇_{τ,[n]} then gives rise to a group morphism B_{1,n} → exp(𝔱̂_{1,n}) ⋊ S_{n}, which factors through B_{1,n}(ℂ) → exp(𝔱̂_{1,n}) ⋊ S_{n}. Since this map commutes with the natural morphisms to S_{n} and using the isomorphism PB_{1,n}(ℂ) ≃ exp(𝔱̂_{1,n}), we obtain that B_{1,n}(ℂ) → exp(𝔱̂_{1,n}) ⋊ S_{n} is an isomorphism.

Similarly, from the exact sequence 1 → \overline{PB}_{1,n} → \overline{B}_{1,n} → S_{n} → 1 one defines a group \overline{B}_{1,n}(ℂ) fitting in an exact sequence 1 → \overline{PB}_{1,n} → \overline{B}_{1,n}(ℂ) → S_{n} → 1 together with an isomorphism \overline{B}_{1,n}(ℂ) → exp(ˆ𝔱̄_{1,n}) ⋊ S_{n}.

3 Bundles with flat connection on M_{1,n} and M_{1,[n]}

We first define Lie algebras of derivations of 𝔱̄_{1,n} and a related group G_{n}. We then define a principal G_{n}-bundle with flat connection of M_{1,n} and a principal G_{n} ⋊ S_{n}-bundle with flat connection on the moduli space M_{1,[n]} of elliptic curves with n unordered marked points.

3.1 Derivations of the Lie algebras 𝔱_{1,n} and 𝔱̄_{1,n} and associated groups

Let 𝔡 be the Lie algebra with generators Δ_{0}, d, X and δ_{2m} (m ≥ 1), and relations:

[d, X] = 2X, [d, Δ_{0}] = −2Δ_{0}, [X, Δ_{0}] = d,
[δ_{2m}, X] = 0, [d, δ_{2m}] = 2mδ_{2m}, ad(Δ_{0})^{2m+1}(δ_{2m}) = 0.

Proposition 7. We have a Lie algebra morphism 𝔡 → Der(𝔱_{1,n}), denoted by ξ ↦ ξ̃, such that d̃(x_{i}) = x_{i}, d̃(y_{i}) = −y_{i}, d̃(t_{ij}) = 0, X̃(x_{i}) = 0, X̃(y_{i}) = x_{i}, X̃(t_{ij}) = 0, Δ̃_{0}(x_{i}) = y_{i}, Δ̃_{0}(y_{i}) = 0, Δ̃_{0}(t_{ij}) = 0, δ̃_{2m}(x_{i}) = 0,

δ̃_{2m}(t_{ij}) = [t_{ij}, (adx_{i})^{2m}(t_{ij})], and
δ̃_{2m}(y_{i}) = ∑_{j|j≠i} \frac{1}{2} ∑_{p+q=2m−1}[(adx_{i})^{p}(t_{ij}), (−adx_{i})^{q}(t_{ij})].

This induces a Lie algebra morphism 𝔡 → Der(𝔱̄_{1,n}).

Proof. The fact that Δ̃_{0}, d̃, X̃ are derivations and commute according to the Lie bracket of 𝔰l_{2} is clear.

Let us prove that δ̃_{2m} is a derivation. We have

δ̃_{2m}(t_{ij}) = [t_{ij}, ∑_{i<j}(adx_{i})^{2m}(t_{ij})],

which implies that δ̃_{2m} preserves the infinitesimal pure braid identities. It clearly preserves the relations

[x_{i}, x_{j}] = 0, [x_{i}, y_{j}] = t_{ij}, [x_{k}, t_{ij}] = 0, [x_{i} + x_{j}, t_{ij}] = 0.

Let us prove that δ̃_{2m} preserves the relation [y_{k}, t_{ij}] = 0. On the one hand

[δ̃_{2m}(y_{k}), t_{ij}] = \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q}[[(adx_{k})^{p}(t_{ki}), (adx_{k})^{q}(t_{ki})]
+[(adx_{k})^{p}(t_{kj}), (adx_{k})^{q}(t_{kj})], t_{ij}]
= \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q+1}[[(adx_{k})^{p}(t_{ki}), (adx_{k})^{q}(t_{kj})]
+[(adx_{k})^{p}(t_{kj}), (adx_{k})^{q}(t_{ki})], t_{ij}]
= ∑_{p+q=2m−1}(−1)^{q+1}[[(adx_{k})^{p}(t_{ki}), (adx_{k})^{q}(t_{kj})], t_{ij}]
= [t_{ij}, ∑_{p+q=2m−1}(−1)^{p}(adx_{i})^{p}(adx_{j})^{q}([t_{ki}, t_{kj}])].

On the other hand

[y_{k}, δ̃_{2m}(t_{ij})] = [y_{k}, [t_{ij}, (adx_{i})^{2m}(t_{ij})]] = [t_{ij}, [y_{k}, (adx_{i})^{2m}(t_{ij})]].

Now

[y_{k}, (adx_{i})^{2m}(t_{ij})] = − ∑_{α+β=2m−1}(adx_{i})^{α}([t_{ki}, (adx_{i})^{β}(t_{ij})])
= − ∑_{α+β=2m−1}(adx_{i})^{α}[t_{ki}, (−adx_{j})^{β}(t_{ij})]
= − ∑_{α+β=2m−1}(adx_{i})^{α}(−adx_{j})^{β}([t_{ki}, t_{ij}])
= ∑_{p+q=2m−1}(−1)^{p+1}(adx_{i})^{p}(adx_{j})^{q}([t_{ki}, t_{kj}]).

Hence [δ̃_{2m}(y_{k}), t_{ij}] + [y_{k}, δ̃_{2m}(t_{ij})] = 0.

Let us prove that δ̃_{2m} preserves the relation [y_{i}, y_{j}] = 0, i.e., that [δ̃_{2m}(y_{i}), y_{j}] + [y_{i}, δ̃_{2m}(y_{j})] = 0. We have

[y_{i}, δ̃_{2m}(y_{j})] = \frac{1}{2}[y_{i}, ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{ji}), (adx_{j})^{q}(t_{ji})]]

+\frac{1}{2} ∑_{k≠i,j}[y_{i}, ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{jk}), (adx_{j})^{q}(t_{jk})]].

Now

\frac{1}{2}[y_{i}, ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{ji}), (adx_{j})^{q}(t_{ji})]] − (i ↔ j)  (5)

= −\frac{1}{2}[y_{i} + y_{j}, ∑_{p+q=2m−1}(−1)^{q}[(adx_{i})^{p}(t_{ij}), (adx_{i})^{q}(t_{ij})]]

= ∑_{p+q=2m−1}(−1)^{q+1}[[y_{i} + y_{j}, (adx_{i})^{p}(t_{ij})], (adx_{i})^{q}(t_{ij})].

A computation similar to the above computation of [y_{k}, (adx_{i})^{2m}(t_{ij})] yields

[y_{i} + y_{j}, (adx_{i})^{p}(t_{ij})] = (−1)^{p} ∑_{α+β=p−1}[(adx_{k})^{α}(t_{ik}), (adx_{j})^{β}(t_{jk})],

so

(5) = ∑_{α+β+γ=2m−2}[(adx_{i})^{α}(t_{ij}), [(adx_{k})^{β}(t_{ik}), (adx_{j})^{γ}(t_{jk})]].

If now k ≠ i, j, then

[y_{i}, \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{jk}), (adx_{j})^{q}(t_{jk})]]

= ∑_{p+q=2m−1}(−1)^{q}[[y_{i}, (adx_{j})^{p}(t_{jk})], (adx_{j})^{q}(t_{jk})].

As we have seen,

[y_{j}, (adx_{i})^{p}(t_{ik})] = (−1)^{p} ∑_{α+β=p−1}(−adx_{i})^{α}(adx_{k})^{β}[t_{ij}, t_{ik}]
= (−1)^{p+1} ∑_{α+β=p−1}[(−adx_{i})^{α}(t_{ij}), (adx_{k})^{β}(t_{jk})].

So we get that [y_{i}, \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{jk}), (adx_{j})^{q}(t_{jk})]] equals

∑_{α+β+γ=2m−2}[[(adx_{i})^{α}(t_{ij}), (adx_{k})^{β}(t_{ik})], (adx_{j})^{γ}(t_{jk})]

and thus [y_{i}, \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q}[(adx_{j})^{p}(t_{jk}), (adx_{j})^{q}(t_{jk})]] − (i ↔ j) equals

∑_{α+β+γ=2m−2}[(adx_{i})^{α}(t_{ij}), [(adx_{k})^{β}(t_{ik}), (adx_{j})^{γ}(t_{jk})]].

Therefore [y_{i}, δ̃_{2m}(y_{j})] + [δ̃_{2m}(y_{i}), y_{j}] = 0.

Since δ̃_{2m}(∑_{i} x_{i}) = δ̃_{2m}(∑_{i} y_{i}) = 0 and ∑_{i} x_{i} and ∑_{i} y_{i} are central, δ̃_{2m} preserves the relations [∑_{i} x_{i}, y_{j}] = 0 and [∑_{k} x_{k}, t_{ij}] = [∑_{k} y_{k}, t_{ij}] = 0. It follows that δ̃_{2m} preserves the relations [x_{i} + x_{j}, t_{ij}] = [y_{i} + y_{j}, t_{ij}] = 0 and [x_{i}, y_{i}] = − ∑_{j|j≠i} t_{ij}. All this proves that δ̃_{2m} is a derivation.

Let us show that ad(Δ̃_{0})^{2m+1}(δ̃_{2m}) = 0 for m ≥ 1. We have

ad(Δ̃_{0})^{2m+1}(δ̃_{2m})(x_{i}) = −(2m + 1)Δ̃_{0}^{2m} ∘ δ̃_{2m} ∘ Δ̃_{0}(x_{i})
= −(2m + 1)Δ̃_{0}^{2m} ∘ δ̃_{2m}(y_{i})
= −(2m + 1)Δ̃_{0}^{2m}(\frac{1}{2} ∑_{\begin{matrix}j|j ≠ i, \\ p + q = 2m − 1 \end{matrix}} [(adx_{i})^{p}(t_{ij}), (−adx_{i})^{q}(t_{ij})])

= 0;

the last part of this computation implies that ad(Δ̃_{0})^{2m+1}(δ̃_{2m})(y_{i}) = 0, therefore ad(Δ̃_{0})^{2m+1}(δ̃_{2m}) = 0.

We have clearly [X̃, δ̃_{2m}] = 0 and [d̃, δ̃_{2m}] = 2mδ̃_{2m}. It follows that we have a Lie algebra morphism 𝔡 → Der(𝔱_{1,n}). Since d̃, Δ̃_{0}, X̃ and δ̃_{2m} all map ℂ(∑_{i} x_{i}) ⊕ ℂ(∑_{i} y_{i}) to itself, this induces a Lie algebra morphism 𝔡 → Der(𝔱̄_{1,n}). ⊓ ⊔

Let e, f, h be the standard basis of 𝔰l_{2}. Then we have a Lie algebra morphism 𝔡 → 𝔰l_{2}, defined by δ_{2n} ↦ 0, d ↦ h, X ↦ e, Δ_{0} ↦ f. We denote by 𝔡_{+} ⊂ 𝔡 its kernel.

Since the morphism 𝔡 → 𝔰l_{2} has a section (given by e, f, h ↦ X, Δ_{0}, d), we have a semidirect product decomposition 𝔡 = 𝔡_{+} ⋊ 𝔰l_{2}.

We then have

𝔱̄_{1,n} ⋊ 𝔡 = (𝔱̄_{1,n} ⋊ 𝔡_{+}) ⋊ 𝔰l_{2}.

Lemma 8. 𝔱̄_{1,n} ⋊ 𝔡_{+} is positively graded.

Proof. We define compatible ℤ^{2}-gradings of 𝔡 and 𝔱̄_{1,n} by deg(Δ_{0}) = (−1, 1), deg(d) = (0, 0), deg(X) = (1, −1), deg(δ_{2m}) = (2m + 1, 1), deg(x_{i}) = (1, 0), deg(y_{i}) = (0, 1), deg(t_{ij}) = (1, 1).

We define the support of 𝔡 (resp., 𝔱̄_{1,n}) as the subset of ℤ^{2} of indices for which the corresponding component of 𝔡 (resp., 𝔱̄_{1,n}) is nonzero.

Since the x̄_{i} on one hand, the ȳ_{i} on the other hand generate abelian Lie subalgebras of 𝔱̄_{1,n}, the support of 𝔱̄_{1,n} is contained in ℕ_{>0}^{2} ∪ {(1, 0), (0, 1)}.

On the other hand, 𝔡_{+} is generated by the ad(Δ_{0})^{p}(δ_{2m}), which all have degrees in ℕ_{>0}^{2}. It follows that the support of 𝔡_{+} is contained in ℕ_{>0}^{2}.

Therefore the support of 𝔱̄_{1,n} ⋊ 𝔡_{+} is contained in ℕ_{>0}^{2} ∪ {(1, 0), (0, 1)}, so this Lie algebra is positively graded. ⊓ ⊔

Lemma 9. 𝔱̄_{1,n} ⋊ 𝔡_{+} is a sum of finite dimensional 𝔰l_{2}-modules; 𝔡_{+} is a sum of irreducible odd dimensional 𝔰l_{2}-modules.

Proof. A generating space for 𝔱̄_{1,n} is ∑_{i}(ℂx̄_{i} ⊕ ℂȳ_{i}), which is a sum of finite dimensional 𝔰l_{2}-modules, so 𝔱̄_{1,n} is a sum of finite dimensional 𝔰l_{2}-modules.

A generating space for 𝔡_{+} is the sum over m ≥ 1 of its 𝔰l_{2}-submodules generated by the δ_{2m}, which are zero or irreducible odd dimensional, therefore 𝔡_{+} is a sum of odd dimensional 𝔰l_{2}-modules. (In fact, the 𝔰l_{2}-submodule generated by δ_{2m} is nonzero, as it follows from the construction of the above morphism 𝔡_{+} → Der(𝔱̄_{1,n}) that δ_{2m} ≠ 0.) ⊓ ⊔

It follows that 𝔱̄_{1,n}, 𝔡̄_{+} and 𝔱̄_{1,n} ⋊ 𝔡_{+} integrate to SL_{2}(ℂ)-modules (while 𝔡̄_{+} even integrates to a PSL_{2}(ℂ)-module).

We can form in particular the semidirect products

G_{n} := exp((𝔱̄_{1,n} ⋊ 𝔡_{+})^{∧}) ⋊ SL_{2}(ℂ)

and exp(𝔡̂_{+}) ⋊ PSL_{2}(ℂ); we have morphisms G_{n} → exp(𝔡̂_{+}) ⋊ PSL_{2}(ℂ) (this is a 2-covering if n = 1 since 𝔱̄_{1,1} = 0).

Observe that the action of S_{n} by automorphisms of 𝔱̄_{1,n} extends to an action on 𝔱̄_{1,n} ⋊ 𝔡, where the action on 𝔡 is trivial. This gives rise to an action of S_{n} by automorphisms of G_{n}.

3.2 Bundle with flat connection on M_{1,n}

The semidirect product ((ℤ^{n})^{2} × ℂ) ⋊ SL_{2}(ℤ) acts on (ℂ^{n} × ℌ) − Diag_{n} by

(n, m, u) ∗ (z, τ) := (n + τm + u(∑_{i} δ_{i}), τ) for (n, m, u) ∈ (ℤ^{n})^{2} × ℂ

and

(\begin{matrix}α & β \\ γ & δ \end{matrix}) ∗ (z, τ) := (\frac{z}{γτ + δ} , \frac{ατ + β}{γτ + δ}) for (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ SL_{2}(ℤ)

(here Diag_{n} := {(z, τ) ∈ ℂ^{n} × ℌ| for some i ≠ j, z_{ij} ∈ Λ_{τ} }). The quotient then identifies with the moduli space M_{1,n} of elliptic curves with n marked points.

Set G_{n} := exp((𝔱̄_{1,n}⋊𝔡_{+})^{∧})⋊SL_{2}(ℂ). We will define a principal G_{n}-bundle with flat connection (P_{n}, ∇_{P_{n}}) over M_{1,n}.

For u ∈ ℂ^{×}, u^{d} := (\begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix}) ∈ SL_{2}(ℂ) ⊂ G_{n} and for v ∈ ℂ, e^{vX} :=

(\begin{matrix}1 & v \\ 0 & 1 \end{matrix}) ∈ SL_{2}(ℂ) ⊂ G_{n}. Since [X, x̄_{i}] = 0, we consistently set

exp(aX + ∑_{i} b_{i}x̄_{i}) := exp(aX)exp(∑_{i} b_{i}x̄_{i}).

Proposition 10. There exists a unique principal G_{n}-bundle P_{n} over M_{1,n}, such that a section of U ⊂ M_{1,n} is a function f : π^{−1}(U) → G_{n} (where

π : (ℂ^{n} × ℌ) − Diag_{n} → M_{1,n}

is the canonical projection), such that

• f(z + δ_{i}|τ) = f(z + u(∑_{i} δ_{i})|τ) = f(z|τ),

• f(z + τδ_{i}|τ) = e^{−2πix̄_{i}}f(z|τ),

• f(z|τ + 1) = f(z|τ) and

• f(\frac{z}{τ} | − \frac{1}{τ}) = τ^{d}exp(\frac{2πi}{τ}(∑_{i} z_{i}x̄_{i} + X))f(z|τ).

Proof. Let cg̃ : ℂ^{n} × ℌ → G_{n} be a family of holomorphic functions (where g̃ ∈ ((ℤ^{n})^{2} × ℂ) ⋊ SL_{2}(ℤ)) satisfying the cocycle condition

cg̃_{}_{g̃′}(z|τ) = cg̃(g̃′ ∗ (z|τ))c_{}_{g̃′}(z|τ).

Then there exists a unique principal G_{n}-bundle over M_{1,n} such that a section of U ⊂ M_{1,n} is a function f : π^{−1}(U) → G_{n} such that f(g̃ ∗ (z|τ)) = cg̃(z|τ)f(z|τ).

We will now prove that there is a unique cocycle such that c_{(u,0,0)} = c_{(0,δ_{i},0)} = 1, c_{(0,0,δ_{i})} = e^{−2πix̄_{i}}, c_{S} = 1 and c_{T} (z|τ) = τ^{d}exp(\frac{2πi}{τ}(∑_{i} z_{i}x̄_{i} + X)),

where S = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}), T = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}).

Such a cocycle is the same as a family of functions c_{g} : ℂ^{n}×ℌ → G_{n} (where g ∈ SL_{2}(ℤ)), satisfying the cocycle conditions c_{gg′}(z|τ) = c_{g}(g′ ∗(z|τ))c_{g′}(z|τ) for g, g′ ∈ SL_{2}(ℤ), and c_{g}(z + δ_{i}|τ) = e^{2πiγx̄_{i}}c_{g}(z|τ), c_{g}(z + τδ_{i}|τ) =

e^{−2πiδx̄_{i}}c_{g}(z|τ)e^{2πix̄_{i}} and c_{g}(z + u(∑_{i} δ_{i})|τ) = c_{g}(z|τ) for g = (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈

SL_{2}(ℤ).

Lemma 11. There exists a unique family of functions c_{g} : ℂ^{n} ×ℌ → G_{n} such that c_{gg′}(z|τ) = c_{g}(g′ ∗ (z|τ))c_{g′}(z|τ) for g, g′ ∈ SL_{2}(ℤ), with

c_{S}(z|τ) = 1, c_{T} (z|τ) = τ^{d}e^{(2πi/τ)(∑_{j} z_{j}x̄_{j}+X)}.

Proof. SL_{2}(ℤ) is the group generated by S̃, T̃ and relations T̃^{4} = 1, (S̃T̃)^{3} = T̃^{2}, S̃T̃^{2} = T̃^{2}S̃. Let ⟨S̃, T̃⟩ be the free group with generators S̃, T̃; then there is a unique family of maps cg̃ : ℂ^{n} ×ℌ → G_{n}, g̃ ∈ ⟨S̃, T̃⟩ satisfying the cocycle conditions (w.r.t. the action of ⟨S̃, T̃⟩ on ℂ^{n} × ℌ through its quotient SL_{2}(ℤ)) and cS̃ = c_{S}, cT̃ = c_{T}. It remains to show that cT̃^{4} = 1, c_{(S̃T̃)^{3}} = cT̃^{2} and c_{S̃T̃^{2}} = c_{T̃^{2}S̃}.

For this, we show that cT̃^{2}(z|τ) = (−1)^{d}. We have

cT̃^{2}(z|τ) = c_{T} (z/τ| − 1/τ)c_{T} (z|τ)
= (−τ)^{−d}exp(−2πiτ(∑_{j}(z_{j}/τ)x̄_{j} + X))τ^{d}exp(\frac{2πi}{τ}(∑_{j} z_{j}x̄_{j} + X))
= (−1)^{d}

since τ^{d}Xτ^{−d} = τ^{2}X, τ^{d}x̄_{i}τ^{−d} = τx̄_{i}.

Since ((−1)^{d})^{2} = 1^{d} = 1, we get cT̃^{4} = 1. Since cS̃ and cT̃^{2} are both constant and commute, we also get c_{S̃T̃^{2}} = c_{T̃^{2}S̃}.

We finally have c_{S̃T̃}(z|τ) = c_{T} (z|τ) while S̃T̃ = (\begin{matrix}1 & −1 \\ 1 & 0 \end{matrix}), (S̃T̃)^{2} = (\begin{matrix}0 & −1 \\ 1 & −1 \end{matrix})

so

c_{(S̃T̃)^{3}}(z|τ) = c_{T} (\frac{z}{τ − 1}|\frac{1}{1 − τ})c_{T} (\frac{z}{τ} |\frac{τ − 1}{τ})c_{T} (z|τ)
= (\frac{1}{1 − τ})^{d}exp(−2πi ∑ z_{j}x̄_{j} + 2πi(1 − τ)X)(\frac{τ − 1}{τ})^{d}

exp(\frac{2πi}{τ − 1} ∑_{j} z_{j}x̄_{j} + 2πi\frac{τ}{τ − 1}X)τ^{d}exp(\frac{2πi}{τ}(∑_{j} z_{j}x̄_{j} + X))

= (−1)^{d}exp(\frac{2πi}{1 − τ} (∑_{j} z_{j}x̄_{j} + X))exp(\frac{2πi}{τ(τ − 1)}(∑_{j} z_{j}x̄_{j} + X))

exp(\frac{2πi}{τ}(∑_{j} z_{j}x̄_{j} + X)) = (−1)^{d},

so c_{(S̃T̃)^{3}} = cT̃^{2}. ⊓ ⊔

End of proof of Proposition 10. We now check that the maps c_{g} satisfy the remaining conditions, i.e., c(z + u(∑_{i} δ_{i})|τ) = c_{g}(z|τ), c_{g}(z + δ_{i}|τ) = e^{2πiγx̄_{i}}c_{g}(z|τ), c_{g}(z + τδ_{i}|τ) = e^{−2πiδx̄_{i}}c_{g}(z|τ)e^{2πix̄_{i}}. The cocycle identity c_{gg′}(z|τ) = c_{g}(g′∗(z|τ))c_{g′}(z|τ) implies that it suffices to prove these identities for g = S and g = T. They are trivially satisfied if g = S. When g = T, the first identity follows from ∑_{i} x̄_{i} = 0, the third identity follows from the fact that (X, x̄_{1}, ..., x̄_{n}) is a commutative family, the second identity follows from the same fact together with τ^{d}x̄_{i}τ^{−d} = τx̄_{i}. ⊓ ⊔

Set

g(z, x|τ) := \frac{θ(z + x|τ)}{θ(z|τ)θ(x|τ)}(\frac{θ′}{θ} (z + x|τ) − \frac{θ′}{θ} (x|τ)) + \frac{1}{x^{2}} = k_{x}(z, x|τ),

(we set f′(z|τ) := (∂/∂z)f(z|τ)).

We have g(z, x|τ) ∈ Hol((ℂ × ℌ) − Diag_{1})[[x]], therefore g(z, adx̄_{i}|τ) is a linear map 𝔱̄_{1,n} → (Hol((ℂ × ℌ) − Diag_{1}) ⊗ 𝔱̄_{1,n})^{∧}, so g(z, adx̄_{i}|τ)(t̄_{ij}) ∈ (Hol((ℂ × ℌ) − Diag_{1}) ⊗ 𝔱̄_{1,n})^{∧}. Therefore

g(z|τ) := ∑_{i<j} g(z_{ij}, adx̄_{i}|τ)(t̄_{ij})

is a meromorphic function ℂ^{n} × ℌ → ˆ 𝔱̄_{1,n} with only poles at Diag_{n}. We set

Δ̄(z|τ) := −\frac{1}{2πi}Δ_{0} − \frac{1}{2πi} ∑_{n≥1} a_{2n}E_{2n+2}(τ)δ_{2n} + \frac{1}{2πi}g(z|τ),

where a_{2n} = −(2n + 1)B_{2n+2}(2iπ)^{2n+2}/(2n + 2)! and B_{n} are the Bernoulli numbers given by x/(e^{x} − 1) = ∑_{r≥0}(B_{r}/r!)x^{r}. This is a meromorphic function ℂ^{n} × ℌ → (𝔱̄_{1,n} ⋊ 𝔡_{+})^{∧} ⋊ 𝔫_{+} ⊂ Lie(G_{1,n}) (where 𝔫_{+} = ℂΔ_{0} ⊂ 𝔰l_{2}) with only poles at Diag_{n}.

For ψ(x) = ∑_{n≥1} b_{2n}x^{2n}, we set δ_{ψ} := ∑_{n≥1} b_{2n}δ_{2n}, Δ_{ψ} := Δ_{0} + ∑_{n≥1} b_{2n}δ_{2n}. If we set

φ(x|τ) = −x^{−2} −(θ′/θ)′(x|τ)+(x^{−2} +(θ′/θ)′(x|τ))_{|x=0} = g(0, 0|τ)−g(0, x|τ),

then φ(x|τ) = ∑_{n≥1} a_{2n}E_{2n+2}(τ)x^{2n}, so that

Δ̄(z|τ) = −\frac{1}{2πi}Δ_{φ(∗|τ)} + \frac{1}{2πi}g(z|τ).

Theorem 12. There is a unique flat connection ∇_{P_{n}} on P_{n}, whose pull-back to (ℂ^{n} × ℌ) − Diag_{n} is the connection

d − Δ̄(z|τ)dτ − ∑_{i} K̄_{i}(z|τ)dz_{i}

on the trivial G_{n}-bundle.

Proof. We should check that the connection d − Δ̄(z|τ)dτ − ∑_{i} K̄_{i}(z|τ)dz_{i} is equivariant and flat, which is expressed as follows (taking into account that we already checked the equivariance and flatness of d − ∑_{i} K̄_{i}(z|τ)dz_{i} for any τ):

(equivariance) for g = (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ SL_{2}(ℤ)

\frac{1}{γτ + δ} K̄_{i}( \frac{z}{γτ + δ} | \frac{ατ + β}{γτ + δ}) = Ad(c_{g}(z|τ))(K̄_{i}(z|τ)) + [(∂/∂z_{i})c_{g}(z|τ)]c_{g}(z|τ)^{−1} (6)

Δ̄(z + δ_{i}|τ) = Δ̄(z + u(∑_{i} δ_{i})|τ) = Δ̄(z|τ)

and Δ̄(z + τδ_{i}|τ) = e^{−2πiadx_{i}}(Δ̄(z|τ) − K̄_{i}(z|τ)),  (7)

\frac{1}{(γτ + δ)^{2}} Δ̄( \frac{z}{γτ + δ} | \frac{ατ + β}{γτ + δ}) = Ad(c_{g}(z|τ))(Δ̄(z|τ))  (8)

+ \frac{γ}{γz + δ} ∑_{i=1}^{n} z_{i}Ad(c_{g}(z|τ))(K̄_{i}(z|τ)) + [( \frac{∂}{∂τ} + \frac{γ}{γτ + δ} ∑_{i=1}^{n} z_{i} \frac{∂}{∂z_{i}} )c_{g}(z|τ)]c_{g}(z|τ)^{−1},

(flatness) [∂/∂τ − Δ̄(z|τ), ∂/∂z_{i} − K̄_{i}(z|τ)] = 0.

Let us now check the equivariance identity (6) for K̄_{i}(z|τ). The cocycle identity c_{gg′}(z|τ) = c_{g}(g′∗(z|τ))c_{g′}(z|τ) implies that it suffices to check it when g = S and g = T. When g = S, this is the identity K̄_{i}(z|τ + 1) = K̄_{i}(z|τ), which follows from the identity θ(z|τ + 1) = θ(z|τ). When g = T, we have to check the identity

\frac{1}{τ} K̄_{i}(\frac{z}{τ} | − \frac{1}{τ}) = Ad(τ^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)})(K̄_{i}(z|τ)) + 2πix̄_{i}.  (9)

We have

2πix̄_{i} − Ad(e^{2πi(∑_{i} z_{i}x̄_{i}+X)})(ȳ_{i}/τ)

= −Ad(e^{2πi(∑_{i} z_{i}x̄_{i})})(ȳ_{i}/τ) (as Ad(e^{2πiτX})(ȳ_{i}/τ) = ȳ_{i}/τ + 2πix̄_{i})
= −\frac{ȳ_{i}}{τ} − \frac{e^{2πiad(∑_{k} z_{k}x̄_{k})} − 1}{ad(∑_{k} z_{k}x̄_{k})}([∑_{j} z_{j}x̄_{j}, \frac{ȳ_{i}}{τ}])

= −\frac{ȳ_{i}}{τ} − \frac{e^{2πiad(∑_{k} z_{k}x̄_{k})} − 1}{ad(∑_{k} z_{k}x̄_{k})}( ∑_{j|j≠i} \frac{z_{ji}}{τ} t̄_{ij})

= −\frac{ȳ_{i}}{τ} − ∑_{j|j≠i} \frac{e^{2πiad(∑_{k} z_{k}x̄_{k})} − 1}{ad(∑_{k} z_{k}x̄_{k})}(\frac{z_{ji}}{τ} t̄_{ij}) = −\frac{ȳ_{i}}{τ} − ∑_{j|j≠i} \frac{e^{2πiad(z_{ij}x̄_{i})} − 1}{ad(z_{ij}x̄_{i})}(\frac{z_{ji}}{τ} t̄_{ij})

= −\frac{ȳ_{i}}{τ} + ∑_{j|j≠i} \frac{e^{2πiad(z_{ij}x̄_{i})} − 1}{ad(x̄_{i})}(\frac{t̄_{ij}}{τ}),

therefore

\frac{1}{τ} (∑_{j} \frac{e^{2πiz_{ij}adx̄_{i}} − 1}{adx̄_{i}}(t̄_{ij}) − ȳ_{i}) = −Ad(τ^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)})(ȳ_{i}) + 2πix̄_{i}. (10)

We have θ(z/τ| − 1/τ) = (1/τ)e^{(πi/τ)z^{2}}θ(z|τ), therefore

\frac{1}{τ} k(\frac{z}{τ} , x| − \frac{1}{τ}) = e^{2πizx}k(z, τx|τ) + \frac{e^{2πizx} − 1}{xτ}.  (11)

Substituting (z, x) = (z_{ij}, adx̄_{i}) (j ≠ i), applying to t̄_{ij}, summing over j and adding up identity (10), we get

\frac{1}{τ} ( ∑_{j|j≠i} k(\frac{z_{ij}}{τ} , adx̄_{i}| − \frac{1}{τ})(t̄_{ij}) − ȳ_{i}) = ∑_{j|j≠i} e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij})

−Ad(τ^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)})(ȳ_{i}) + 2πix̄_{i}.

Since e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij}) = Ad(τ^{d}e^{(2πi/τ)(∑_{i} z_{i}x̄_{i}+X)})(k(z_{ij}, adx̄_{i})(t̄_{ij})), this implies (9). This ends the proof of (6).

Let us now check the shift identities (7) in Δ̄(z|τ). The first part is immediate; let us check the last identity. We have k(z + τ, x|τ) = e^{−2πix}g(z, x|τ) + (e^{−2πix} − 1)/x, therefore g(z + τ, x|τ) = e^{−2πix}g(z, x|τ) − 2πie^{−2πix}k(z, x|τ) + \frac{1}{x}(\frac{1−e^{−2πix}}{x} − 2πie^{−2πix}). Substituting (z, x) = (z_{ij}, adx̄_{i}) (j ≠ i), applying to t̄_{ij}, summing up and adding up ∑_{k,l|k,l≠j} g(z_{kl}, adx̄_{k}|τ)(t̄_{kl}), we get that g(z + τδ_{i}|τ) equals

e^{−2πiadx̄_{i}}(g(z|τ)) − 2πie^{−2πiadx̄_{i}}(K̄_{i}(z|τ) + ȳ_{i}) + ∑_{j|j≠i} \frac{1}{adx̄_{i}} ( \frac{1 − e^{−2πiadx̄_{i}}}{adx̄_{i}} − 2πie^{−2πiadx̄_{i}})(t̄_{ij})

= e^{−2πiadx̄_{i}}(g(z|τ)) − 2πie^{−2πiadx̄_{i}}(K̄_{i}(z|τ) + ȳ_{i}) − ( \frac{1 − e^{−2πiadx̄_{i}}}{adx̄_{i}} − 2πie^{−2πiadx̄_{i}})(ȳ_{i})

= e^{−2πiadx̄_{i}}(g(z|τ)) − 2πie^{−2πiadx̄_{i}}(K̄_{i}(z|τ)) − \frac{1 − e^{−2πiadx̄_{i}}}{adx̄_{i}} (ȳ_{i});

on the other hand, we have e^{−2πiadx̄_{i}}(Δ_{0}) = Δ_{0}+ \frac{1−e^{−2πiadx̄_{i}}}{adx̄_{i}} (ȳ_{i}) (as [Δ_{0}, x̄_{i}] = ȳ_{i}), therefore g(z + δ_{i}|τ) − Δ_{0} = e^{−2πiadx̄_{i}}(g(z|τ) − Δ_{0} − 2πiK̄_{i}(z|τ)). Since the δ_{2n} commute with x̄_{i}, we get Δ̄(z + τδ_{i}|τ) = e^{−2πiadx̄_{i}}(Δ̄(z|τ) − K̄_{i}(z|τ)), as wanted.

Let us now check the equivariance identities (8) for Δ̄(z|τ). As above, the cocycle identities imply that it suffices to check (8) for g = S, T. When g = S, this identity follows from ∑_{i} K̄_{i}(z|τ) = 0. When g = T, it is written

\frac{1}{τ^{2}} Δ̄(\frac{z}{τ} | − \frac{1}{τ}) = Ad(c_{T}(z|τ))(Δ̄(z|τ) + \frac{1}{τ} ∑_{i} z_{i}K̄_{i}(z|τ)) + \frac{d}{τ} − 2πiX. (12)

The modularity identity (11) for k(z, x|τ) implies that

\frac{1}{τ^{2}} g(\frac{z}{τ} , x|−\frac{1}{τ}) = e^{2πizx}g(z, τx|τ)+\frac{2πiz}{τ}e^{2πizx}k(z, τx|τ)+\frac{1 − e^{2πizx}}{τ^{2}x^{2}} +\frac{2πiz}{τ^{2}} \frac{e^{2πizx}}{x} .

This implies

\frac{1}{τ^{2}} ∑_{i<j} g(\frac{z_{ij}}{τ} , adx̄_{i}| − \frac{1}{τ})(t̄_{ij}) = ∑_{i<j} e^{2πiz_{ij}adx̄_{i}}g(z_{ij}, τadx̄_{i}|τ)(t̄_{ij})

+ ∑_{i<j} \frac{2πi}{τ}z_{ij}e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij})+∑_{i<j}(\frac{1 − e^{2πiz_{ij}adx̄_{i}}}{τ^{2}(adx̄_{i})^{2}} +\frac{2πiz_{ij}}{τ^{2}} \frac{e^{2πiz_{ij}adx̄_{i}}}{adx̄_{i}} )(t̄_{ij}).

We compute as above

∑_{i<j} e^{2πiz_{ij}adx̄_{i}}g(z_{ij}, τadx̄_{i}|τ)(t̄_{ij}) = Ad(τ^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)})(g(z|τ)),

∑_{i<j} \frac{2πi}{τ}z_{ij}e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij}) = ∑_{i} \frac{2πi}{τ}z_{i}(∑_{j|j≠i} e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij}))

(using k(z, x|τ) + k(−z, −x|τ) = 0) and

∑_{i<j} e^{2πiz_{ij}adx̄_{i}}k(z_{ij}, τadx̄_{i}|τ)(t̄_{ij}) = Ad(τ^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)})(K̄_{i}(z|τ) + ȳ_{i}).

Therefore

\frac{1}{τ^{2}} g(\frac{z}{τ} | − \frac{1}{τ}) = Ad(c_{T}(z|τ))(g(z|τ) + \frac{2πi}{τ} ∑_{i} z_{i}K̄_{i}(z|τ) + \frac{2πi}{τ} ∑_{i} z_{i}ȳ_{i})

+ ∑_{i<j}(\frac{1 − e^{2πiz_{ij}adx̄_{i}}}{τ^{2}(adx̄_{i})^{2}} + \frac{2πiz_{ij}}{τ^{2}} \frac{e^{2πiz_{ij}adx̄_{i}}}{adx̄_{i}} )(t̄_{ij}),

which implies

\frac{1}{τ^{2}} Δ̄(\frac{z}{τ} | − \frac{1}{τ}) = Ad(c_{T}(z|τ))(Δ̄(z|τ) + \frac{1}{τ} ∑_{i} K̄_{i}(z|τ)) + Ad(c_{T}(z|τ))(\frac{1}{τ} ∑_{i} z_{i}ȳ_{i})

+\frac{1}{2πi} ∑_{i<j}(\frac{1 − e^{2πiz_{ij}adx̄_{i}}}{τ^{2}(adx̄_{i})^{2}} + \frac{2πiz_{ij}}{τ^{2}} \frac{e^{2πiz_{ij}adx̄_{i}}}{adx̄_{i}} )(t̄_{ij})

+\frac{1}{2πi}(Ad(c_{T}(z|τ))(Δ_{φ(∗|τ)}) − \frac{1}{τ^{2}} Δ_{φ(∗|−1/τ)}).

To prove (12), it then suffices to prove

Ad(c_{T}(z|τ))(\frac{1}{τ} ∑_{i} z_{i}ȳ_{i}) + \frac{1}{2πi} ∑_{i<j}(\frac{1 − e^{2πiz_{ij}adx̄_{i}}}{τ^{2}(adx̄_{i})^{2}} + \frac{2πiz_{ij}}{τ^{2}} \frac{e^{2πiz_{ij}adx̄_{i}}}{adx̄_{i}} )(t̄_{ij})

+\frac{1}{2πi}(Ad(c_{T}(z|τ))(Δ_{φ(∗|τ)}) − \frac{1}{τ^{2}} Δ_{φ(∗|−1/τ)}) = \frac{d}{τ} − 2πiX.  (13)

We compute

Ad(c_{T}(z|τ))(\frac{1}{τ} ∑_{i} z_{i}ȳ_{i}) = \frac{1}{τ^{2}} ∑_{i} z_{i}ȳ_{i}+\frac{2πi}{τ} ∑_{i} z_{i}x̄_{i}+∑_{i<j}(−\frac{1}{τ^{2}} )z_{ij} \frac{e^{2πiz_{ij}adx̄_{i}} − 1}{adx̄_{i}}(t̄_{ij}).

We also have Ad(c_{T}(z|τ))(E_{2n+2}(τ)δ_{2n}) = \frac{1}{τ^{2}} E_{2n+2}(−\frac{1}{τ})δ_{2n} since [δ_{2n}, x̄_{i}] = [δ_{2n}, X] = 0 and [d, δ_{2n}] = 2nδ_{2n}, and since E_{2n+2}(−1/τ) = τ^{2n+2}E_{2n+2}(τ). This implies

Ad(c_{T}(z|τ))(δ_{φ(∗|τ)}) = δ_{φ(∗|−1/τ)}.

We now compute Ad(c_{T}(z|τ))(Δ_{0}) − (1/τ^{2})Δ_{0}. We have

Ad(c_{T}(z|τ))(Δ_{0}) = Ad(e^{2πi ∑_{i} z_{i}x̄_{i}}) ∘ Ad(τ^{d}e^{(2πi/τ)X})(Δ_{0}),

and

Ad(τ^{d}e^{(2πi/τ)X})(Δ_{0}) = (1/τ^{2})Δ_{0} + (2πi/τ)d − (2πi)^{2}X.

Now Ad(e^{2πi ∑_{i} z_{i}x̄_{i}})(X) = X, Ad(e^{2πi ∑_{i} z_{i}x̄_{i}})(d) = d − 2πi ∑_{i} z_{i}x̄_{i}. We now compute

Ad(e^{2πi ∑_{i} z_{i}x̄_{i}})(Δ_{0}) = Δ_{0} + \frac{e^{2πi ∑_{i} z_{i}adx̄_{i}} − 1}{2πiad(∑_{i} z_{i}x̄_{i})}([2πi ∑_{i} z_{i}x̄_{i}, Δ_{0}])

= Δ_{0} − \frac{e^{2πi ∑_{i} z_{i}adx̄_{i}} − 1}{ad(∑_{i} z_{i}x̄_{i})}(∑_{i} z_{i}ȳ_{i}) = Δ_{0} − ∑_{i} \frac{e^{2πi ∑_{j|j≠i} z_{ji}adx̄_{j}} − 1}{ad(∑_{j|j≠i} z_{ji}x̄_{j})}(z_{i}ȳ_{i})

= Δ_{0} − ∑_{i} (2πiz_{i}ȳ_{i} + \frac{1}{ad(∑_{j|j≠i} z_{ji}x̄_{j})}(\frac{e^{2πi ∑_{j|j≠i} z_{ji}adx̄_{j}} − 1}{ad(∑_{j|j≠i} z_{ji}x̄_{j})} − 2πi)([∑_{j|j≠i} z_{ji}x̄_{j}, z_{i}ȳ_{i}]))

= Δ_{0} − ∑_{i} 2πiz_{i}ȳ_{i} − ∑_{i≠j} (\frac{1}{ad(x̄_{j})}(\frac{e^{2πiz_{ji}adx̄_{j}} − 1}{ad(z_{ji}x̄_{j})} − 2πi)(z_{i}t̄_{ij})); the last sum decomposes as

∑_{i<j} \frac{1}{ad(x̄_{j})}(\frac{e^{2πiz_{ji}adx̄_{j}} − 1}{ad(z_{ji}x̄_{j})} − 2πi)(z_{i}t̄_{ij}) + ∑_{i>j} \frac{1}{ad(x̄_{j})}(\frac{e^{2πiz_{ji}adx̄_{j}} − 1}{ad(z_{ji}x̄_{j})} − 2πi)(z_{i}t̄_{ij})

= ∑_{i<j} \frac{1}{ad(x̄_{j})}(\frac{e^{2πiz_{ji}adx̄_{j}} − 1}{ad(z_{ji}x̄_{j})} − 2πi)(z_{i}t̄_{ij}) + \frac{1}{ad(x̄_{i})}(\frac{e^{2πiz_{ij}adx̄_{i}} − 1}{ad(z_{ij}x̄_{i})} − 2πi)(z_{j}t̄_{ij})

= ∑_{i<j} \frac{1}{ad(x̄_{i})}(\frac{e^{2πiz_{ij}adx̄_{i}} − 1}{ad(z_{ij}x̄_{i})} − 2πi)(z_{ji}t̄_{ij}),

so

Ad(e^{2πi ∑_{i} z_{i}x̄_{i}})(Δ_{0}) = Δ_{0}−2πi ∑_{i} z_{i}ȳ_{i}−∑_{i<j} \frac{1}{ad(x̄_{i})}(\frac{e^{2πiz_{ij}adx̄_{i}} − 1}{ad(z_{ij}x̄_{i})} −2πi)(z_{ji}t̄_{ij}),

and finally

Ad(c_{T}(z|τ))(Δ_{φ(∗|τ)}) − \frac{1}{τ^{2}} Δ_{φ(∗|−1/τ)} = −\frac{2πi}{τ^{2}} ∑_{i} z_{i}ȳ_{i} −

\frac{1}{τ^{2}} ∑_{i<j} \frac{1}{ad(x̄_{i})}(\frac{e^{2πiz_{ij}adx̄_{i}} − 1}{ad(z_{ij}x̄_{i})} − 2πi)(z_{ji}t̄_{ij}) + \frac{2πi}{τ}(d − 2πi ∑_{i} z_{i}x̄_{i}) − (2πi)^{2}X,

which implies (13). This proves (12) and therefore (8).

We prove the flatness identity [∂/∂τ − Δ̄(z|τ), ∂/∂z_{i} − K̄_{i}(z|τ)] = 0. For this, we prove that (∂/∂τ)K̄_{i}(z|τ) = (∂/∂τ)Δ̄(z|τ), and [Δ̄(z|τ), K̄_{i}(z|τ)] = 0.

Let us first prove

(∂/∂τ)K̄_{i}(z|τ) = (∂/∂z_{i})Δ̄(z|τ).  (14)

We have

(∂/∂τ)K̄_{i}(z|τ) = ∑_{j|j≠i}(∂_{τ} k)(z_{ij}, adx̄_{i}|τ)(t̄_{ij})

and (∂/∂z_{i})Δ̄(z|τ) = (2πi)^{−1} ∑_{j|j≠i} (∂_{z}g)(z_{ij}, adx̄_{i})(t̄_{ij}) (where ∂_{τ} := ∂/∂τ, ∂_{z} = ∂/∂z) so it suffices to prove the identity

(∂_{τ} k)(z, x|τ) = (2πi)^{−1}(∂_{z}g)(z, x|τ),

i.e., (∂_{τ} k)(z, x|τ) = (2πi)^{−1}(∂_{z}∂_{x}k)(z, x|τ). In this identity, k(z, x|τ) may be replaced by k̃(z, x|τ) := k(z, x|τ) + 1/x = θ(z + x|τ)/(θ(z|τ)θ(x|τ)). Dividing by k̃(z, x|τ), the wanted identity is rewritten as

2πi(\frac{∂_{τ} θ}{θ} (z + x|τ) − \frac{∂_{τ} θ}{θ} (z|τ) − \frac{∂_{τ} θ}{θ} (x|τ))

= (\frac{θ′}{θ})′(z + x|τ) + (\frac{θ′}{θ} (z + x|τ) − \frac{θ′}{θ} (z|τ))(\frac{θ′}{θ} (z + x|τ) − \frac{θ′}{θ} (x|τ))

(recall that f′(z|τ) = ∂_{z}f(z|τ)), or taking into account the heat equation 4πi(∂_{τ} θ/θ)(z|τ) = (θ′′/θ)(z|τ) − 12πi(∂_{τ} η/η)(τ), as follows

2(\frac{θ′}{θ} (z|τ)\frac{θ′}{θ} (x|τ) − \frac{θ′}{θ} (x|τ)\frac{θ′}{θ} (z + x|τ) − \frac{θ′}{θ} (z|τ)\frac{θ′}{θ} (z + x|τ))  (15)
+\frac{θ′′}{θ} (z|τ) + \frac{θ′′}{θ} (x|τ) + \frac{θ′′}{θ} (z + x|τ) − 12πi\frac{∂_{τ} η}{η} (τ) = 0

Let us prove (15). Denote its l.h.s. by F (z, x|τ). Since θ(z|τ) is odd w.r.t. z, F (z, x|τ) is invariant under the permutation of z, x, −z − x. The identities

(θ′/θ)(z + τ|τ) = (θ′/θ)(z|τ) − 2πi

and

(θ′′/θ)(z + τ|τ) = (θ′′/θ)(z|τ) − 4πi(θ′/θ)(z|τ) + (2πi)^{2}

imply that F (z, x|τ) is elliptic in z, x (w.r.t. the lattice Λ_{τ}). The possible poles of F (z, x|τ) as a function of z are simple at z = 0 and z = −x (mod Λ_{τ}), but one checks that F (z, x|τ) is regular at these points, so it is constant in z. By the 𝔖_{3}-symmetry, it is also constant in x, hence it is a function of τ only: F (z, x|τ) = F (τ).

To compute this function, we compute

F (z, 0|τ) = [−2(θ′/θ)′ −2(θ′/θ)^{2} +2θ′′/θ](z|τ)+(θ′′/θ)(0|τ)−12πi(∂_{τ} η/θ)(τ),

hence

F (τ) = (θ′′/θ)(0|τ) − 12πi(∂_{τ} η/η)(τ).

The above heat equation then implies that F (τ) = 4πi(∂_{τ} θ/θ)(0|τ). Now θ′(0|τ) = 1 implies that θ(z|τ) has the expansion θ(z|τ) = z + ∑_{n≥2} a_{n}(τ)z^{n} as z → 0, which implies (∂_{τ} θ/θ)(0|τ) = 0. So F (τ) = 0, which implies (15) and therefore (14).

We now prove

[Δ̄(z|τ), K̄_{i}(z|τ)] = 0.  (16)

Since τ is constant in what follows, we will write k(z, x), g(z, x), φ instead of k(z, x|τ), g(z, x|τ), φ(∗|τ). For i ≠ j, let us set g_{ij} := g(z_{ij}, adx̄_{i})(t̄_{ij}). Since g(z, x|τ) = g(−z, −x|τ), we have g_{ij} = g_{ji}. Recall that K̄_{ij} = k(z_{ij}, adx̄_{i})(t_{ij}).

We have

2πi[Δ̄(z|τ), K̄_{i}(z|τ)] = [−Δ_{φ} + ∑_{i,j|i<j} g_{ij}, −ȳ_{i} + ∑_{j|j≠i} K̄_{ij}]  (17)

= [Δ_{φ}, ȳ_{i}] + ∑_{j|j≠i} ( − [Δ_{φ}, K̄_{ij}] + [ȳ_{i}, g_{ij}] + [g_{ij}, K̄_{ij}])

+ ∑_{j,k|j≠i,k≠i,j<k} ([ȳ_{i}, g_{jk}] + [g_{ik} + g_{jk}, K̄_{ij}] + [g_{ij} + g_{jk}, K̄_{ik}]).

One computes

[Δ_{φ}, ȳ_{i}] = ∑_{α}[f_{α}(adx̄_{i})(t̄_{ij}), g_{α}(−adx̄_{i})(t̄_{ij})]  (18)

where ∑_{α} f_{α}(u)g_{α}(v) = \frac{1}{2} \frac{φ(u)−φ(v)}{u−v}. If f(x) ∈ ℂ[[x]], then

[Δ_{0}, f(adx̄_{i})(t̄_{ij})] − [ȳ_{i}, f′(adx̄_{i})(t̄_{ij})] = ∑_{α}[h_{α}(adx̄_{i})(t̄_{ij}), k_{α}(adx̄_{i})(t̄_{ij})]

+ ∑_{k|k≠i,j} \frac{f(adx̄_{i}) − f(−adx̄_{j}) − f′(−adx̄_{j})(adx̄_{i} + adx̄_{j})}{(adx̄_{i} + adx̄_{j})^{2}}([t̄_{ij}, t̄_{ik}]),

where

∑_{α} h_{α}(u)k_{α}(v) = \frac{1}{2}(\frac{1}{v^{2}} (f(u+v)−f(u)−vf′(u))−\frac{1}{u^{2}} (f(u+v)−f(v)−uf′(v))).

Since g(z, x) = k_{x}(z, x), we get

−[Δ_{0}, K̄_{ij}] + [ȳ_{i}, g_{ij}] = − ∑_{α}[f_{α}^{ij}(adx̄_{i})(t̄_{ij}), g_{α}^{ij}(adx̄_{i})(t̄_{ij})]  (19)

+ ∑_{k|k≠i,j} \frac{k(z_{ij}, adx̄_{i}) − k(z_{ij}, −adx̄_{j}) − (adx̄_{i} + adx̄_{j})k_{x}(z_{ij}, −adx̄_{j})}{(adx̄_{i} + adx̄_{j})^{2}}([t̄_{ij}, t̄_{jk}]),

where ∑_{α} f_{α}^{ij}(u)g_{α}^{ij}(v) equals

\frac{1}{2}(\frac{1}{v^{2}} (k(z_{ij}, u+v)−k(z_{ij}, u)−vk_{x}(z_{ij}, u))−\frac{1}{u^{2}} (k(z_{ij}, u+v)−k(z_{ij}, v)−uk_{x}(z_{ij}, v))).

For f(x) ∈ ℂ[[x]], we have

[δ_{φ}, f(adx̄_{i})(t̄_{ij})] = ∑_{α}[l_{α}(adx̄_{i})(t̄_{ij}), m_{α}(adx̄_{i})(t̄_{ij})],

where ∑_{α} l_{α}(u)m_{α}(v) = f(u + v)φ(v); therefore

−[δ_{φ}, K̄_{ij}] = − ∑_{α}[l_{α}^{ij}(adx̄_{i})(t̄_{ij}), m_{α}^{ij}(adx̄_{i})(t̄_{ij})],  (20)

where ∑_{α} l_{α}^{ij}(u)m_{α}^{ij}(v) = k(z_{ij}, u + v)φ(v).

For j, k ≠ i and j < k, we have

[ȳ_{i}, g_{jk}] + [g_{ik} + g_{jk}, K̄_{ij}] + [g_{ij} + g_{jk}, K̄_{ik}]
= [ȳ_{i}, g_{jk}] − [g_{ki}, K̄_{ji}] − [g_{ji}, K̄_{ki}] + [g_{jk}, K̄_{ij}] + [g_{jk}, K̄_{ik}],

and since for any f(x) ∈ ℂ[[x]], [ȳ_{i}, f(adx̄_{i})(t̄_{jk})] = −\frac{f(adx̄_{j})−f(−adx̄_{k})}{adx̄_{j}+adx̄_{k}}([t̄_{ij}, t̄_{jk}]), we get

[ȳ_{i}, g_{jk}] + [g_{ik} + g_{jk}, K̄_{ij}] + [g_{ij} + g_{jk}, K̄_{ik}] = ( − \frac{g(z_{jk}, adx̄_{j}) − g(z_{jk}, −adx̄_{k})}{adx̄_{j} + adx̄_{k}}

−g(z_{ki}, adx̄_{k})k(z_{ji}, adx̄_{j}) + g(z_{ji}, adx̄_{j})k(z_{ki}, adx̄_{k})  (21)

−g(z_{kj}, adx̄_{k})k(z_{ij}, adx̄_{i}) + g(z_{jk}, adx̄_{j})k(z_{ik}, adx̄_{i}))([t̄_{ij}, t̄_{jk}]).

Summing up (18), (19), (20) and (21), (17) gives

2πi[Δ̄(z|τ), K̄_{i}(z|τ)] = ∑_{j|j≠i} ∑_{α}[F_{α}^{ij}(adx̄_{i})(t̄_{ij}), G_{α}^{ij}(adx̄_{i})(t̄_{ij})]

+ ∑_{j,k|j≠i,k≠i} H(z_{ij}, z_{ik}, −adx̄_{j}, −adx̄_{k})([t_{ij}, t_{jk}]),

where ∑_{α} F_{α}^{ij}(u)G_{α}^{ij}(v) = L(z_{ij}, u, v),

L(z, u, v) = \frac{1}{2} \frac{φ(u) − φ(v)}{u + v} + \frac{1}{2} k(z, u + v)(φ(u) − φ(v))

+ \frac{1}{2} (g(z, u)k(z, v) − k(z, u)g(z, v))

− \frac{1}{2} ( \frac{1}{v^{2}} (k(z, u + v) − k(z, u) − vk_{x}(z, u))

− \frac{1}{u^{2}} (k(z, u + v) − k(z, v) − uk_{x}(z, v)))

and

H(z, z′, u, v) =  \frac{1}{v^{2}} (k(z, u + v) − k(z, u) − vk_{x}(z, u))
− \frac{1}{u^{2}} (k(z′, u + v) − k(z′, v) − uk_{x}(z′, v))
+ \frac{1}{u+v} (g(z′ − z, −u) − g(z′ − z, v))
−g(−z′, −v)k(−z, −u) + g(−z, −u)k(−z′, −v)
−g(z − z′, −v)k(z, u + v) + g(z′ − z, −u)k(z′, u + v).

Explicit computation shows that H(z, z′, u, v) = 0, which implies that L(z, u, v) = 0 since L(z, u, v) = −\frac{1}{2}H(z, z, u, v). This proves (16). ⊓ ⊔

Remark 13. Define Δ(z|τ) by the same formula as Δ̄(z|τ), replacing x̄_{i}, ȳ_{i} by x_{i}, y_{i}. Then d − Δ(z|τ)dτ − ∑_{i} K_{i}(z|τ)dz_{i} is flat. This can be interpreted as follows.

Let N_{+} ⊂ SL_{2}(ℂ) be the connected subgroup with Lie algebra ℂΔ_{0}. Set Ñ_{n} := exp((𝔱_{1,n} ⋊ 𝔡_{+})^{∧}) ⋊ N_{+}, N_{n} := exp((𝔱̄_{1,n} ⋊ 𝔡_{+})^{∧}) ⋊ N_{+} and G̃_{n} := exp((𝔱_{1,n} ⋊ 𝔡_{+})^{∧}) ⋊ SL_{2}(ℂ). Then we have a diagram of groups

(\begin{matrix}Ñ_{n} & → & N_{n} \\ ↓ & & ↓ \\ G̃_{n} & → & G_{n} \end{matrix})

The trivial N_{n}-bundle on (ℌ×ℂ^{n})−Diag_{n} with flat connection d−Δ̄(z|τ)dτ − ∑_{i} K̄_{i}(z|τ)dz_{i} admits a reduction to Ñ_{n}, where the bundle is again trivial and the connection is d − Δ(z|τ)dτ − ∑_{i} K_{i}(z|τ)dz_{i}.

((ℤ^{2})^{2} × ℂ) ⋊ SL_{2}(ℤ) contains the subgroups (ℤ^{n})^{2}, (ℤ^{n})^{2} × ℂ, (ℤ^{n})^{2} ⋊ SL_{2}(ℤ). We denote the corresponding quotients of (ℂ^{n} × ℌ) − Diag_{n} by C(n), C̄(n), M̃_{1,n}. These fit in the diagram

(\begin{matrix}C̃(n) & → & C(n) \\ ↓ & & ↓ \\ M̃_{1,n} & → & M_{1,n} \end{matrix})

The pair (P_{n}, ∇_{P_{n}}) can be pulled back to G_{n}-bundles over these covers of M_{1,n}. These pull-backs admit G-structures, where G is the corresponding group in the above diagram of groups.

We have natural projections C(n) → ℌ, C̄(n) → ℌ. The fibers of τ ∈ ℌ are respectively C(E_{τ} , n) and C̄(E_{τ} , n). The pair (P_{n}, ∇_{n}) can be pulled back to C(E_{τ} , n) and C̄(E_{τ} , n); these pull-backs admit G-structures, where G = exp(𝔱_{1,n}) and exp(𝔱̄_{1,n}), which coincide with (P_{n,τ} , ∇_{n,τ}) and (P̄_{n,τ} , ¯∇_{n,τ}).

3.3 Bundle with flat connection over M_{1,[n]}

The semidirect product ((ℤ^{n})^{2} × ℂ) ⋊ (SL_{2}(ℤ) × S_{n}) acts on (ℂ^{n} × ℌ) − Diag_{n} as follows: the action of ((ℤ^{n})^{2} × ℂ) ⋊ SL_{2}(ℂ) is as above and the action of S_{n} is σ ∗ (z_{1}, ..., z_{n}, τ) := (z_{σ^{−1}(1)}, ..., z_{σ^{−1}(n)}, τ). The quotient then identifies with M_{1,[n]}.

We will define a principal G_{n}⋊S_{n}-bundle with a flat connection (P_{[n]}, ∇_{P_{[n]}}) over M_{1,[n]}.

Proposition 14. There exists a unique principal G_{n} ⋊ S_{n}-bundle P_{[n]} over M_{1,[n]}, such that a section of U ⊂ M_{1,[n]} is a function f : π̃^{−1}(U) → G_{n}⋊S_{n}, satisfying the conditions of Proposition 10 as well as f(σz|τ) = σf(z|τ) for σ ∈ S_{n} (here π̃ : (ℂ^{n} × ℌ) − Diag_{n} → M_{1,[n]} is the canonical projection).

Proof. One checks that σcg̃(z|τ)σ^{−1} = c_{σg̃σ^{−1}}(σ^{−1}z), where g̃ ∈ ((ℤ^{n})^{2} ×ℂ)⋊ SL_{2}(ℤ), σ ∈ S_{n}. It follows that there is a unique cocycle c_{(g̃,σ)} : ℂ^{n} × ℌ → G_{n} ⋊ S_{n} such that c_{(g̃,1)} = cg̃ and c_{(1,σ)}(z|τ) = σ. ⊓ ⊔

Theorem 15. There is a unique flat connection ∇_{P_{[n]}} on P_{[n]}, whose pullback to (ℂ^{n} × ℌ) − Diag_{n} is the connection d − Δ̄(z|τ)dτ − ∑_{i} K̄_{i}(z|τ)dz_{i} on the trivial G_{n} ⋊ S_{n}-bundle.

Proof. Taking into account Theorem 12, it remains to show that this connection is S_{n}-equivariant. We have already mentioned that ∑_{i} K̄_{i}(z|τ)dz_{i} is equivariant; Δ̄(z|τ) is also checked to be equivariant. ⊓ ⊔

4 The monodromy morphisms Γ_{1,[n]} → G_{n} ⋊ S_{n}

Let Γ_{1,[n]} be the mapping class group of genus 1 surfaces with n unordered marked points. It can be viewed as the fundamental group π_{1}(M_{1,[n]}, ∗), where ∗ is a base point at infinity which will be specified later. The flat connection on M_{1,[n]} introduced above gives rise to morphisms γ_{n} : Γ_{1,[n]} → G_{n} ⋊ S_{n}, which we now study. This study in divided in two parts: in the first, analytic part, we show that γ_{n} can be obtained from γ_{1} and γ_{2}, and show that the restriction of γ_{n} to \overline{B}_{1,n} can be expressed in terms of the KZ associator only. In the second part, we show that morphisms \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n}) ⋊ S_{n} can be constructed algebraically using an arbitrary associator. Finally, we introduce the notion of an elliptic structure over a quasi-bialgebra.

4.1 The solution F^{(n)}(z|τ)

The elliptic KZB system is now

(∂/∂z_{i})F (z|τ) = K̄_{i}(z|τ)F (z|τ), (∂/∂τ)F (z|τ) = Δ̄(z|τ)F (z|τ),

where F (z|τ) is a function (ℂ^{n} × ℌ) − Diag_{n} ⊃ U → G_{n} ⋊ S_{n} invariant under translation by ℂ(∑_{i} δ_{i}). Let D_{n} := {(z, τ) ∈ ℂ^{n} × ℌ|z_{i} = a_{i} + b_{i}τ, a_{i}, b_{i} ∈ ℝ, a_{1} < a_{2} < ... < a_{n} < a_{1} + 1, b_{1} < b_{2} < ... < b_{n} < b_{1} + 1}. Then D_{n} ⊂ (ℂ^{n} × ℌ)−Diag_{n} is simply connected and invariant under ℂ(∑_{i} δ_{i}). A solution of the elliptic KZB system on this domain is then unique, up to right multiplication by a constant. We now determine a particular solution F^{(n)}(z|τ).

Let us study the elliptic KZB system in the region z_{ij} ≪ 1, τ → i∞. Then K̄_{i}(z|τ) = ∑_{j|j≠i} t̄_{ij}/(z_{i} − z_{j}) + O(1).

We now compute the expansion of Δ̄(z|τ). The heat equation for ϑ implies the expansion ϑ(x|τ) = η(τ)^{3}(x + 2πi∂_{τ} logη(τ)x^{3} + O(x^{5})), so θ(x|τ) = x + 2πi∂_{τ} logη(τ)x^{3} + O(x^{5}), hence

g(0, x|τ) = (\frac{θ′}{θ})′(x|τ) + \frac{1}{x^{2}} = 4πi∂_{τ} logη(τ) + O(x) = −(π^{2}/3)E_{2}(τ) + O(x)

since E_{2}(τ) = \frac{24}{2πi}∂_{τ} logη(τ). We have g(0, x|τ) = g(0, 0|τ) − φ(x|τ), so

g(0, x|τ) = − ∑_{k≥0} a_{2k}x^{2k}E_{2k+2}(τ),

where a_{0} = π^{2}/3. Then

Δ̄(z|τ) = −\frac{1}{2πi}(Δ_{0} + ∑_{k≥0} a_{2k}E_{2k+2}(τ)(δ_{2k} + ∑_{i,j|i<j}(adx̄_{i})^{2k}(t̄_{ij}))) + o(1)

for z_{ij} ≪ 1 and any τ ∈ ℌ. Since we have an expansion E_{2k}(τ) = 1 + ∑_{l>0} a_{kl}e^{2πilτ} as τ → i∞, and using Proposition 85 with u_{n} = z_{n1}, u_{n−1} = z_{n−1,1}/z_{n1},..., u_{2} = z_{21}/z_{31}, u_{1} = q = e^{2πiτ}, there is a unique solution F^{(n)}(z|τ) with the expansion

F^{(n)}(z|τ) ≃ zt̄_{12} ; 21z_{31}^{t̄_{13}+t̄_{23}}...z_{n1}^{t̄_{1n}+...+t̄_{n−1,n}}exp(−\frac{τ}{2πi}(Δ_{0}+∑_{k≥0} a_{2k}(δ_{2k}+∑_{i<j}(adx̄_{i})^{2k}(t̄_{ij}))))

in the region z_{21} ≪ z_{31} ≪ ... ≪ z_{n1} ≪ 1, τ → i∞, (z, τ) ∈ D_{n} (here z_{ij} = z_{i} − z_{j}); here the sign ≃ means that any of the ratios of both sides has the form 1 + ∑_{k>0} ∑_{i,a_{1},...,a_{n}} r_{k}^{i,a_{1},...,a_{n}}(u_{1}, ..., u_{n}), where the second sum is finite with a_{i} ≥ 0, i ∈ {1, ..., n}, r_{k}^{i,a_{1},...,a_{n}}(u_{1}, ..., u_{n}) has degree k, and is O(u_{i}(logu_{1})^{a_{1}}...(logu_{n})^{a_{n}}).

4.2 Presentation of Γ_{1,[n]}

According to [Bir69b], Γ_{1,[n]} = \{\overline{B}_{1,n} ⋊ { \atop SL_{2}(ℤ)}\}/ℤ, where { \atop SL_{2}(ℤ)} is a

central extension 1 → ℤ → { \atop SL_{2}(ℤ)} → SL_{2}(ℤ) → 1; the action α : { \atop SL_{2}(ℤ)} → Aut(\overline{B}_{1,n}) is such that for Z the central element 1 ∈ ℤ ⊂ { \atop SL_{2}(ℤ)}, α_{Z}(x) = Z′x(Z′)^{−1}, where Z′ is the image of a generator of the center of PB_{n} (the pure braid group of n points on the plane) under the natural morphism PB_{n} → \overline{B}_{1,n}; \overline{B}_{1,n} ⋊ { \atop SL_{2}(ℤ)} is then \overline{B}_{1,n} × { \atop SL_{2}(ℤ)} with the product (p, A)(p′, A′) = (pα_{A}(p′), AA′); this semidirect product is then factored by its central subgroup (isomorphic to ℤ) generated by ((Z′)^{−1}, Z).

Γ_{1,[n]} is presented explicitly as follows. Generators are σ_{i} (i = 1, ..., n − 1), A_{i}, B_{i} (i = 1, ..., n), C_{jk} (1 ≤ j < k ≤ n), Θ and Ψ, and relations are:

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} (i = 1, ..., n − 2), σ_{i}σ_{j} = σ_{j}σ_{i} (1 ≤ i < j ≤ n),

σ_{i}^{−1}X_{i}σ_{i}^{−1} = X_{i+1}, σ_{i}Y_{i}σ_{i} = Y_{i+1} (i = 1, ..., n − 1),

(σ_{i}, X_{j}) = (σ_{i}, Y_{j}) = 1 (i ∈ {1, ..., n − 1}, j ∈ {1, ..., n}, j ≠ i, i + 1),

σ_{i}^{2} = C_{i,i+1}C_{i+1,i+2}C_{i,i+2}^{−1} (i = 1, ..., n − 1),
(A_{i}, A_{j}) = (B_{i}, B_{j}) = 1(any i, j), A_{1} = B_{1} = 1,
(B_{k}, A_{k}A_{j}^{−1}) = (B_{k}B_{j}^{−1}, A_{k}) = C_{jk} (1 ≤ j < k ≤ n),
(A_{i}, C_{jk}) = (B_{i}, C_{jk}) = 1 (1 ≤ i ≤ j < k ≤ n),
ΘA_{i}Θ^{−1} = B_{i}^{−1}, ΘB_{i}Θ^{−1} = B_{i}A_{i}B_{i}^{−1},
ΨA_{i}Ψ^{−1} = A_{i}, ΨB_{i}Ψ^{−1} = B_{i}A_{i}, (Θ, σ_{i}) = (Ψ, σ_{i}) = 1,
(Ψ, Θ^{2}) = 1, (ΘΨ)^{3} = Θ^{4} = C_{12}...C_{n−1,n}.

Here X_{i} = A_{i}A_{i+1}^{−1}, Y_{i} = B_{i}B_{i+1}^{−1} for i = 1, ..., n (with the convention A_{n+1} = B_{n+1} = C_{i,n+1} = 1). The relations imply

C_{jk} = σ_{j,j+1...k}...σ_{j+n−k,j+n−k+1...n}σ_{j,j+1...n−k+j+1}...σ_{k−1,k...n},

where σ_{i,i+1...j} = σ_{j−1}...σ_{i}. Observe that C_{12}, ..., C_{n−1,n} commute with each other.

The group { \atop SL_{2}(ℤ)} is presented by generators Θ, Ψ and Z, and relations: Z is central, Θ^{4} = (ΘΨ)^{3} = Z and (Ψ, Θ^{2}) = 1. The morphism { \atop SL_{2}(ℤ)} → SL_{2}(ℤ)

is Θ ↦ ( (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ), Ψ ↦ ( (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) ), and the morphism Γ_{1,[n]} → SL_{2}(ℤ) is

given by the same formulas and A_{i}, B_{i}, σ_{i} ↦ 1.

The elliptic braid group \overline{B}_{1,n} is the kernel of Γ_{1,[n]} → SL_{2}(ℤ); it has the same presentation as Γ_{1,[n]}, except for the omission of the generators Θ, Ψ and the relations involving them. The “pure” mapping class group Γ_{1,n} is the kernel of Γ_{1,[n]} → S_{n}, A_{i}, B_{i}, C_{jk} ↦ 1, σ_{i} ↦ σ_{i}; it has the same presentation as Γ_{1,[n]}, except for the omission of the σ_{i}. Finally, recall that \overline{PB}_{1,n} is the kernel of Γ_{1,[n]} → SL_{2}(ℤ) × S_{n}.

Remark 16. The extended mapping class group Γ̃_{1,n} of classes of non necessarily orientation-preserving self-homeomorphisms of a surface of type (1, n) fits in a split exact sequence 1 → Γ_{1,n} → Γ̃_{1,n} → ℤ/2ℤ → 1; it may be

viewed as \{\overline{PB}_{1,n} ⋊ { \atop GL_{2}(ℤ)}\}/ℤ; it has the same presentation as Γ_{1,n} with the additional generator Σ subject to

Σ^{2} = 1, ΣΘΣ^{−1} = Θ^{−1}, ΣΨΣ^{−1} = Ψ^{−1},
ΣA_{i}Σ^{−1} = A_{i}^{−1}, ΣB_{i}Σ^{−1} = A_{i}B_{i}A_{i}^{−1}.

4.3 The monodromy morphisms γ_{n} : Γ_{1,[n]} → G_{n} ⋊ S_{n}

Let F (z|τ) be a solution of the elliptic KZB system defined on D_{n}.

Recall that D_{n} := {(z, τ) ∈ ℂ^{n} × ℌ|z_{i} = a_{i} + b_{i}τ, a_{i}, b_{i} ∈ ℝ, a_{1} < a_{2} < ... < a_{n} < a_{1} + 1, b_{1} < b_{2} < ... < b_{n} < b_{1} + 1}. The domains H_{n} := {(z, τ) ∈ ℂ^{n} × ℌ|z_{i} = a_{i} + b_{i}τ, a_{i}, b_{i} ∈ ℝ, a_{1} < a_{2} < ... < a_{n} < a_{1} + 1} and D_{n} := {(z, τ) ∈ ℂ^{n} ×ℌ|z_{i} = a_{i} +b_{i}τ, a_{i}, b_{i} ∈ ℝ, b_{1} < b_{2} < ... < b_{n} < b_{1} +1} are also simply connected and invariant, and we denote by F^{H}(z|τ) and F^{V} (z|τ) the prolongations of F (z|τ) to these domains.

Then (z, τ) ↦ F^{H}(z + ∑_{j=i}^{n} δ_{i}|τ) and (z, τ) ↦ e^{2πi(x̄_{i}+...+x̄_{n})}F^{V} (z + τ(∑_{j=i}^{n} δ_{i})|τ) are solutions of the elliptic KZB system on H_{n} and D_{n} respectively. We define A_{i}^{F} , B_{i}^{F} ∈ G_{n} by

F^{H}(z+∑_{j=i}^{n} δ_{i}|τ) = F^{H}(z|τ)A_{i}^{F} , e^{2πi(x̄_{i}+...+x̄_{n})}F^{V} (z+τ(∑_{j=i}^{n} δ_{i})|τ) = F^{V} (z|τ)B_{i}^{F} .

The action of T^{−1} = ( (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ) is (z, τ) ↦ (−z/τ, −1/τ); this trans-

formation takes H_{n} to V_{n}. Then (z, τ) ↦ c_{T^{−1}}(z|τ)^{−1}F^{V} (−z/τ| − 1/τ) is a solution of the elliptic KZB system on H_{n} (recall that c_{T^{−1}}(z|τ)^{−1} = e^{2πi(− ∑_{i} z_{i}x̄_{i}+τX)}(−τ)^{d} = (−τ)^{d}e^{(2πi/τ)(∑_{i} z_{i}x̄_{i}+X)}). We define Θ^{F} by

c_{T^{−1}}(z|τ)^{−1}F^{V} (−z/τ| − 1/τ) = F^{H}(z|τ)Θ^{F} .

The action of S = ( (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) ) is (z, τ) ↦ (z, τ + 1). This transformation

takes H_{n} to itself. Since c_{S}(z|τ) = 1, the function (z, τ) ↦ F^{H}(z, τ + 1) is a solution of the elliptic KZB system on H_{n}. We define Ψ^{F} by

F^{H}(z|τ + 1) = F^{H}(z|τ)Ψ^{F} .

Finally, define σ_{i}^{F} by

σ_{i}F (σ_{i}^{−1}z|τ) = F (z|τ)σ_{i}^{F} ,

where on the l.h.s. F is extended to the universal cover of (ℂ^{n} × ℌ) − Diag_{n} (σ_{i} exchanges z_{i} and z_{i+1}, z_{i+1} passing to the right of z_{i}).

Lemma 17. There is a unique morphism Γ_{1,[n]} → G_{1,n} ⋊ S_{n}, taking X to X^{F}, where X = A_{i}, B_{i}, Θ or Ψ.

Proof. This follows from the geometric description of generators of Γ_{1,[n]}: if (z_{0}, τ_{0}) ∈ D_{n}, then A_{i} is the class of the projection of the path [0, 1] ∋ t ↦ (z_{0} + t ∑_{j=i}^{n} δ_{j}, τ_{0}), B_{i} is the class of the projection of [0, 1] ∋ t ↦ (z_{0} + tτ ∑_{j=i}^{n} δ_{j}, τ_{0}), Θ is the class of the projection of any path connecting (z_{0}, τ_{0}) to (−z_{0}/τ_{0}, −1/τ_{0}) contained in H_{n}, and Ψ is the class of the projection of any path connecting (z_{0}, τ_{0}) to (z_{0}, τ_{0} + 1) contained in H_{n}. ⊓ ⊔

We will denote by γ_{n} : Γ_{1,[n]} → G_{n} ⋊ S_{n} the morphism induced by the solution F^{(n)}(z|τ).

4.4 Expression of γ_{n} : Γ_{1,[n]} → G_{n} ⋊ S_{n} using γ_{1} and γ_{2}

Lemma 18. There exists a unique Lie algebra morphism 𝔡 → 𝔱̄_{1,n}⋊𝔡, x ↦ [x], such that [δ_{2n}] = δ_{2n} + ∑_{i<j}(adx̄_{i})^{2n}(t̄_{ij}), [X] = X, [Δ_{0}] = Δ_{0}, [d] = d.

It induces a group morphism G_{1} → G_{n}, also denoted g ↦ [g].

Lemma 19. For each map ϕ : {1, ..., m} → {1, ..., n}, there exists a Lie algebra morphism 𝔱̄_{1,n} → 𝔱̄_{1,m}, x ↦ x^{ϕ}, defined by (x̄_{i})^{ϕ} := ∑_{i′∈ϕ^{−1}(i)} x̄_{i′}, (ȳ_{i})^{ϕ} := ∑_{i′∈ϕ^{−1}(i)} ȳ_{i′}, (t̄_{ij})^{ϕ} := ∑_{i′∈ϕ^{−1}(i),j′∈ϕ^{−1}(j)} t̄_{i′j′}.

It induces a group morphism exp(ˆ𝔱̄_{1,n}) → exp(ˆ𝔱̄_{1,m}), also denoted g ↦ g^{ϕ}.

The proofs are immediate. We now recall the definition and properties of the KZ associator [Dri91].

If k is a field with char(k) = 0, we let 𝔱_{n}^{k} be the k-Lie algebra generated by t_{ij}, where i ≠ j ∈ {1, ..., n}, with relations

t_{ji} = t_{ij}, [t_{ij} + t_{ik}, t_{jk}] = 0, [t_{ij}, t_{kl}] = 0

for i, j, k, l distinct (in this section, we set 𝔱_{n} := 𝔱_{n}^{ℂ}). For each partially defined map {1, ..., m} ⊃ D_{ϕ} →^{ϕ} {1, ..., n}, we have a Lie algebra morphism 𝔱_{n} → 𝔱_{m}, x ↦ x^{ϕ}, defined by^{6} (t_{ij})^{ϕ} := ∑_{i′∈ϕ^{−1}(i),j′∈ϕ^{−1}(j)} t_{i′j′}. We also have morphisms 𝔱_{n} → 𝔱_{1,n}, t_{ij} ↦ t̄_{ij}, compatible with the maps x ↦ x^{ϕ} on both sides.

The KZ associator Φ = Φ(t_{12}, t_{23}) ∈ exp(𝔱̂_{3}) is defined by G_{0}(z) = G_{1}(z)Φ, where G_{i} :]0, 1[→ exp(𝔱̂_{3}) are the solutions of G′(z)G(z)^{−1} = t_{12}/z+t_{23}/(z−1) with G_{0}(z) ∼ z^{t_{12}} as z → 0 and G_{1}(z) ∼ (1−z)^{t_{23}} as z → 1. The KZ associator satisfies the duality, hexagon and pentagon equation (37), (38) below (where λ = 2πi).

Lemma 20. γ_{2}(A_{2}) and γ_{2}(B_{2}) belong to exp(ˆ𝔱̄_{1,2}) ⊂ G_{2}.

Proof. If F (z|τ) : H_{2} → G_{2} is a solution of the KZB equation for n = 2, then A_{2}^{F} = F^{H}(z + δ_{2}|τ)F^{H}(z|τ)^{−1} is expressed as the iterated integral, from z_{0} ∈ D_{n} to z_{0} + δ_{2}, of K̄_{2}(z|τ) ∈ ˆ 𝔱̄_{1,2}, hence A_{2}^{F} ∈ exp(ˆ𝔱̄_{1,2}). Since γ_{2}(A_{2}) is a conjugate of A_{2}^{F}, it belongs to exp(ˆ𝔱̄_{1,2}) as exp(ˆ𝔱̄_{1,2}) ⊂ G_{2} ⋊ S_{2} is normal. One proves similarly that γ_{2}(B_{2}) ∈ exp(ˆ𝔱̄_{1,2}). ⊓ ⊔

Set

Φ_{i} := Φ^{1...i−1,i,i+1...n}...Φ^{1...n−2,n−1,n} ∈ exp(𝔱̂_{n}).

We denote by x ↦ {x} the morphism exp(𝔱̂_{n}) → exp(ˆ𝔱̄_{1,n}) induced by t_{ij} ↦ t̄_{ij}.

Proposition 21. If n ≥ 2, then

γ_{n}(Θ) = [γ_{1}(Θ)]e^{i \frac{π}{2} ∑_{i<j} t̄_{ij}}, γ_{n}(Ψ) = [γ_{1}(Ψ)]e^{i \frac{π}{6} ∑_{i<j} t̄_{ij}},

and if n ≥ 3, then

γ_{n}(A_{i}) = {Φ_{i}}^{−1}γ_{2}(A_{2})^{1...i−1,i...n}{Φ_{i}}, (i = 1, ..., n),
γ_{n}(B_{i}) = {Φ_{i}}^{−1}γ_{2}(B_{2})^{1...i−1,i...n}{Φ_{i}}, (i = 1, ..., n),
γ_{n}(σ_{i}) = {Φ^{1...i−1,i,i+1}}^{−1}e^{iπt̄_{i,i+1}}{Φ^{1...i−1,i,i+1}}, (i = 1, ..., n − 1).

Proof. In the region z_{21} ≪ z_{31} ≪ ... ≪ z_{n1} ≪ 1, (z, τ) ∈ D_{n}, we have

F^{(n)}(z|τ) ≃ zt̄_{12} ; 21...z_{n1}^{t̄_{1n}+...+t̄_{n−1,n}}exp(−\frac{a_{0}}{2πi}(∫_{i}^{τ} E_{2} + C)(∑_{i<j} t̄_{ij}))[F(τ)],

where F (τ) = F^{(1)}(z|τ) for any z. Here C is the constant such that ∫_{i}^{τ} E_{2} + C = τ + o(1) as τ → i∞.

We have F (τ + 1) = F (τ)γ_{1}(Ψ), F (−1/τ) = F (τ)γ_{1}(Θ). Since ∑_{i<j} t̄_{ij} commutes with the image of x ↦ [x], we get

F^{(n)}(z|τ + 1) = F^{(n)}(z|τ)exp(−\frac{a_{0}}{2πi}(∑_{i<j} t̄_{ij}))[γ_{1}(Ψ)],

6We will also use the notation x^{I_{1},...,I_{n}} for x^{ϕ}, where I_{i} = ϕ^{−1}(i).

so

γ_{n}(Ψ) = exp(i\frac{π}{6} ∑_{i<j} t̄_{ij})[γ_{1}(Ψ)].

In the same region,

c_{T^{−1}}(z|τ)^{−1}F^{(n)V} (− \frac{z}{τ} | − \frac{1}{τ}) ≃ (−τ)^{d}e^{\frac{2πi}{τ} (∑_{i} z_{i}x̄_{i}+X)}(−z_{21}/τ)t̄_{12}...(−z_{n1}/τ)^{t̄_{1n}+...+t̄_{n−1,n}}

exp(− \frac{a_{0}}{2πi} (∫_{i}^{−1/τ} E_{2} + C)(∑_{i<j} t̄_{ij}))[F(−1/τ)].

Now E_{2}(−1/τ) = τ^{2}E_{2}(τ) + (6i/π)τ, so

∫_{i}^{−1/τ} E_{2} − ∫_{i}^{τ} E_{2} = (6i/π)[log(−1/τ) − logi]

(where log(re^{iθ}) = logr + iθ for θ ∈] − π, π[).

It follows that

c_{T^{−1}}(z|τ)^{−1}F^{(n)V} (− \frac{z}{τ} | − \frac{1}{τ}) ≃ e^{2πi(∑_{i} z_{i}x̄_{i})}zt̄_{12} ; 21...z_{n1}^{t̄_{1n}+...+t̄_{n−1,n}}

exp(− \frac{a_{0}}{2πi} (∫_{i}^{τ} E_{2} + C)(∑_{i<j} t̄_{ij}))(exp − \frac{a_{0}}{2πi} \frac{−6i}{π} (logi)(∑_{i<j} t̄_{ij}))[(−τ)^{d}e^{(2πi/τ)X}F(−1/τ)]

≃ zt̄_{12} ; 21...z_{n1}^{t̄_{1n}+...+t̄_{n−1,n}}exp(− \frac{a_{0}}{2πi} (∫_{i}^{τ} E_{2} + C)(∑_{i<j} t̄_{ij}))[F(τ)γ_{1}(Θ)]exp( \frac{iπ}{2} ∑_{i<j} t̄_{ij})

≃ F^{(n)H}(z|τ)[γ_{1}(Θ)]exp( \frac{iπ}{2} ∑_{i<j} t̄_{ij})

(the second ≃ follows from ∑_{i} z_{i}x̄_{i} = ∑_{i>1} z_{i1}x̄_{i} and z_{i1} → 0), so

γ_{n}(Θ) = [γ_{1}(Θ)]exp(i\frac{π}{2} ∑_{i<j} t̄_{ij}).

Let G_{i}(z|τ) be the solution of the elliptic KZB system, such that

G_{i}(z|τ) ≃ zt̄_{12} ; 21...z_{i−1,1}^{t̄_{12}+...+t̄_{1,i−1}}z_{n,i}^{t̄_{i,n}+...+t̄_{n−1,n}}...zt̄_{n−1,n} ; n,n−1

exp( − \frac{τ}{2πi}(Δ_{0} + ∑_{n≥0} a_{2n}(δ_{2n} + ∑_{i<j}(adx̄_{i})^{2n}(t̄_{ij}))))

when z_{21} ≪ ... ≪ z_{i−1,1} ≪ 1, z_{n,n−1} ≪ ... ≪ z_{n,i} ≪ 1, τ → i∞ and (z, τ) ∈ D_{n}. Then G_{i}(z + ∑_{j=i}^{n} δ_{i}|τ) = G_{i}(z|τ)γ_{2}(A_{2})^{1...i−1,i...n}, because in the domain considered K̄_{i}(z|τ) is close to K̄_{2}(z_{1}, z_{n}|τ)^{1...i−1,i...n} (where K̄_{2}(...) corresponds to the 2-point system); on the other hand, F (z|τ) = G_{i}(z|τ){Φ_{i}}, which implies the formula for γ_{n}(A_{i}). The formula for γ_{n}(B_{i}) is proved in the same way. Finally, the behavior of F^{(n)}(z|τ) for z_{21} ≪ ... ≪ z_{n1} ≪ 1 is similar to that of a solution of the KZ equations, which implies the formula for γ_{n}(σ_{i}). ⊓ ⊔ Remark 22. One checks that the composition SL_{2}(ℤ) ≃ Γ_{1,1} → G_{1} → SL_{2}(ℂ) is a conjugation of the canonical inclusion. It follows that the composition { \atop SL_{2}(ℤ)} ⊂ Γ_{1,n} → G_{1} → SL_{2}(ℂ) is a conjugation of the canonical projection for any n ≥ 1.

Let us set Ã := γ_{2}(A_{2}), B̃ := γ_{2}(B_{2}). The image of A_{2}A_{3}^{−1} = σ_{1}^{−1}A_{2}^{−1}σ_{1}^{−1} by γ_{3} yields

Ã^{12,3} = e^{iπt̄_{12}}{Φ}^{3,1,2}Ã^{2,13}{Φ}^{2,1,3}e^{iπt̄_{12}} · {Φ}^{3,2,1}Ã^{1,23}{Φ}^{1,2,3}  (22)

and the image of B_{2}B_{3}^{−1} = σ_{1}B_{2}^{−1}σ_{1} yields

B̃^{12,3} = e^{−iπt̄_{12}}{Φ}^{3,1,2}B̃^{2,13}{Φ}^{2,1,3}e^{−iπt̄_{12}} · {Φ}^{3,2,1}B̃^{1,23}{Φ}^{1,2,3}. (23)

Since (γ_{3}(A_{2}), γ_{3}(A_{3})) = (γ_{3}(B_{2}), γ_{3}(B_{3})) = 1, we get

({Φ}^{3,2,1}Ã^{1,23}{Φ}, Ã^{12,3}) = ({Φ}^{3,2,1}B̃^{1,23}{Φ}, B̃^{12,3}) = 1  (24)

(this equation can also be directly derived from (22) and (23) by noting that the l.h.s. is invariant x ↦ x^{2,1,3} and commutes with e^{±iπt̄_{12}}). We have for n = 2, C_{12} = (B_{2}, A_{2}), so (Ã, B̃) = γ_{2}(C_{12})^{−1}. Also γ_{1}(Θ)^{4} = 1, so γ_{2}(C_{12}) = γ_{2}(Θ)^{4} = (e^{iπt̄_{12}/2}[γ_{1}(Θ)])^{4} = e^{2πit̄_{12}}[γ_{1}(Θ)^{4}] = e^{2πit̄_{12}}, so

(Ã, B̃) = e^{−2πit̄_{12}}.  (25)

For n = 3, we have γ_{3}(Θ)^{4} = e^{2πi(t̄_{12}+t̄_{13}+t̄_{23})} = γ_{3}(C_{12}C_{23}); since γ_{3}(C_{12}) = (γ_{3}(B_{2}), γ_{3}(A_{2})) = {Φ}^{−1}(B̃, Ã)^{1,23}{Φ} = {Φ}^{−1}e^{2πi(t̄_{12}+t̄_{13})}{Φ}, we get γ_{3}(C_{23}) = {Φ}^{−1}e^{2πit̄_{23}}{Φ}. The image by γ_{3} of (B_{3}, A_{3}A_{2}^{−1}) = (B_{3}B_{2}^{−1}, A_{3}) = C_{23} then gives

(B̃^{12,3}, Ã^{12,3}{Φ}^{−1}(Ã^{1,23})^{−1}{Φ}) = (B̃^{12,3}{Φ}^{−1}(B̃^{1,23})^{−1}{Φ}, Ã^{12,3})
= {Φ}^{−1}e^{2πit̄_{23}}{Φ}  (26)

(applying x ↦ x^{∅,1,2}, this identity implies (25)).

Let us set Θ̃ := γ_{1}(Θ), Ψ̃ := γ_{1}(Θ). Since γ_{1}, γ_{2} are group morphisms, we have

Θ̃^{4} = (Θ̃Ψ̃)^{3} = (Θ̃^{2}, Ψ̃) = 1,  (27)
[Θ̃]e^{i \frac{π}{2} t̄_{12}} Ã([Θ̃]e^{i \frac{π}{2} t̄_{12}})^{−1} = B̃^{−1}, [Θ̃]e^{i \frac{π}{2} t̄_{12}} B̃([Θ̃]e^{i \frac{π}{2} t̄_{12}})^{−1} = B̃ÃB̃^{−1}, (28)
[Ψ̃]e^{i \frac{π}{6} t̄_{12}} Ã([Ψ̃]e^{i \frac{π}{6} t̄_{12}})^{−1} = Ã, [Ψ̃]e^{i \frac{π}{6} t̄_{12}} B̃([Ψ̃]e^{i \frac{π}{6} t̄_{12}})^{−1} = B̃Ã.  (29)

(27) (resp., (28), (29)) are identities in G_{1} (resp., G_{2}); in (28), (29), x ↦ [x] is induced by the map 𝔡 → 𝔡 ⋊ 𝔱̄_{1,2} defined above.

4.5 Expression of Ψ̃ and of Ã and B̃ in terms of Φ

In this section, we compute Ã and B̃ in terms of the KZ associator Φ. We also compute Ψ̃.

Recall the definition of Ψ̃. The elliptic KZB system for n = 1 is

2πi∂_{τ} F(τ) + (Δ_{0} + ∑_{k≥1} a_{2k}E_{2k+2}(τ)δ_{2k})F(τ) = 0.

The solution F (τ) := F^{(1)}(z|τ) (for any z) is determined by F (τ) ≃ exp(−\frac{τ}{2πi}(Δ_{0} + ∑_{k≥1} a_{2k}δ_{2k})). Then Ψ̃ is determined by F (τ + 1) = F (τ)Ψ̃. We have therefore:

Lemma 23. Ψ̃ = exp(−\frac{1}{2πi}(Δ_{0} + ∑_{k≥1} a_{2k}δ_{2k})).

Recall the definition of Ã and B̃. The elliptic KZB system for n = 2 is

∂_{z}F (z|τ) = −(\frac{θ(z + adx|τ)adx}{θ(z|τ)θ(adx|τ)})(y) · F (z|τ),  (30)

2πi∂_{τ} F(z|τ) + (Δ_{0} + ∑_{k≥1} a_{2k}E_{2k+2}(τ)δ_{2k} − g(z, adx|τ)(t))F(z|τ) = 0, (31)

where z = z_{21}, x = x̄_{2} = −x̄_{1}, y = ȳ_{2} = −ȳ_{1}, t = t̄_{12} = −[x, y].

The solution F (z|τ) := F^{(2)}(z_{1}, z_{2}|τ) is determined by its behavior F (z|τ) ≃ z^{t}exp(−\frac{τ}{2πi}(Δ_{0} + ∑_{k≥0} a_{2k}(δ_{2k} + (adx)^{2k})(t))) when z → 0^{+}, τ → i∞. We then have F^{H}(z + 1|τ) = F^{H}(z|τ)Ã, e^{2πix}F^{V} (z + τ|τ) = F^{V} (z|τ)B̃.

Proposition 24. We have^{7}

Ã = (2π/i)^{t}Φ(ỹ, t)e^{2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t}
= (2π)^{t}i^{−3t}Φ(−ỹ − t, t)e^{2πi(ỹ+t)}Φ(−ỹ − t, t)^{−1}(2πi)^{−t},

where ỹ = −\frac{adx}{e^{2πiadx}−1}(y).

Proof. Ã = F^{H}(z|τ)^{−1}F^{H}(z + 1|τ), which we will compute in the limit τ → i∞. For this, we will compute F (z|τ) in the limit τ → i∞. In this limit, θ(z|τ) = (1/π)sin(πz)[1 + O(e^{2πiτ})] so the system becomes

∂_{z}F (z|τ) = (πcotg(πz)t − πcotg(πadx)adx(y) + O(e^{2πiτ}))F(z|τ) (32)

2πi∂_{τ} F(z|τ)+(Δ_{0}+∑_{k≥1} a_{2k}δ_{2k}+(\frac{π^{2}}{sin^{2}(πadx)}−\frac{1}{(adx)^{2}} )(t)+O(e^{2πiτ}))F(z|τ) = 0

where the last equation is

7By convention, if z ∈ ℂ \ ℝ_{−} and x ∈ 𝔫, where 𝔫 is a pronilpotent Lie algebra, then z^{x} is exp(x log z) ∈ exp(𝔫), where log z is chosen with imaginary part in ]−π, π[.

2πi∂_{τ} F(z|τ) + (Δ_{0} + a_{0}t + ∑_{k≥1} a_{2k}(δ_{2k} + (adx)^{2k}(t)) + O(e^{2πiτ}))F(z|τ) = 0.

We set

Δ := Δ_{0} + ∑_{k≥1} a_{2k}δ_{2k}, so Δ_{0} + a_{0}t + ∑_{k≥1} a_{2k}(δ_{2k} + (adx)^{2k}(t)) = [Δ] + a_{0}t.

The compatibility of this system implies that [Δ] + a_{0}t commutes with t and (πadx)cotg(πadx)(y) = iπ(−t − 2ỹ), hence with t and ỹ; actually t commutes with each [δ_{2k}] = δ_{2k} + (adx)^{2k}(t).

Equation (30) can be written ∂_{z}F (z|τ) = (t/z + O(1))F (z|τ). We then let F_{0}(z|τ) be the solution of (30) in V := {(z, τ)|τ ∈ ℌ, z = a + bτ, a ∈ ]0, 1[, b ∈ ℝ} such that F_{0}(z|τ) ≃ z^{t} when z → 0^{+}, for any τ. This means that the left (equivalently, right) ratio of these quantities has the form 1 + ∑_{k>0}(degree k)O(z(logz)^{f(k)}) where f(k) ≥ 0.

We now relate F (z|τ) and F_{0}(z|τ). Let F (τ) = F^{(1)}(z|τ) for any z be the solution of the KZB system for n = 1, such that F (τ) ≃ exp(−\frac{τ}{2πi}Δ) as τ → i∞ (meaning that the left, or equivalently right, ratio of these quantities has the form 1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}), where f(k) ≥ 0).

Lemma 25. We have F (z|τ) = F_{0}(z|τ)exp(−\frac{a_{0}}{2πi}(∫_{i}^{τ} E_{2} + C)t)[F(τ)], where C is such that ∫_{i}^{τ} E_{2} + C = τ + O(e^{2πiτ}).

Proof of Lemma. F (z|τ) = F_{0}(z|τ)X(τ), where X : ℌ → G_{2} is a map. We have g(z, adx|τ)(t) = a_{0}E_{2}(τ)t + ∑_{k>0} a_{2k}E_{2k+2}(τ)(adx)^{2k}(t) + O(z) when z → 0^{+} and for any τ, so (31) is written as

2πi∂_{τ} F(z|τ) + (Δ_{0} + a_{0}E_{2}(τ)t + ∑_{k>0} a_{2k}E_{2k+2}(τ)[δ_{2k}] + O(z))F(z|τ) = 0

where O(z) has degree > 0. Since Δ_{0}, t and the [δ_{2k}] all commute with t, the ratio F_{0}(z|τ)^{−1}F (z|τ) satisfies

2πi∂_{τ} (F_{0}^{−1}F(z|τ)) + (Δ_{0} + a_{0}E_{2}(τ)t + ∑_{k>0} a_{2k}E_{2k+2}(τ)[δ_{2k}]
+ ∑_{k>0}(degree k)O(z(logz)^{h(k)}))(F_{0}^{−1}F(z|τ)) = 0

where h(k) ≥ 0. Since F_{0}(z|τ)^{−1}F (z|τ) = X(τ) is in fact independent on z, we have

2πi∂_{τ} (X(τ)) + (Δ_{0} + a_{0}E_{2}(τ)t + ∑_{k>0} a_{2k}E_{2k+2}(τ)[δ_{2k}])(X(τ)) = 0,

which implies that X(τ) = exp(−\frac{a_{0}}{2πi}(∫_{i}^{τ} E_{2} + C)t)[F(τ)]X_{0}, where X_{0} is a suitable element in G_{2}. The asymptotic behavior of F (z|τ) when τ → i∞ and z → 0^{+} then implies X_{0} = 1. ⊓ ⊔

End of proof of Proposition. We then have F (z|τ) = F_{0}(z|τ)X(τ), where X(τ) ≃ exp(−\frac{τ}{2πi}([Δ] + a_{0}t)) as τ → i∞, where this means that the left ratio (equivalently, the right ratio) of these quantities has the form 1 + ∑_{k>0}(degree k)O(τ^{x(k)}e^{2πiτ}), where x(k) ≥ 0.

If we set u := e^{2πiz}, then (30) is rewritten as

∂_{u}F̄(u|τ) = (ỹ/u + t/(u − 1) + O(e^{2πiτ}))F̄(u|τ),  (33)

where F̄(u|τ) = F (z|τ).

Let D′ := {u||u| ≤ 1} − [0, 1] be the complement of the unit interval in the unit disc. Then we have a bijection {(z, τ)|τ ∈ iℝ_{+}^{×}, z = a + τb, a ∈ [0, 1], b ≥ 0} → D′ × iℝ_{+}^{×}, given by (z, τ) ↦ (u, τ) := (e^{2πiz}, τ).

Let F̄_{a}, F̄_{f} be the solutions of (33) in D′ × iℝ_{+}, such that F̄_{a}(u|τ) ≃ ((u − 1)/(2πi))^{t} when u = 1+i0^{+}, and for any τ, and F̄_{f}(u|τ) ≃ e^{iπt}((1−u)/(2πi))^{t} when u = 1 − i0^{+}, for any τ.

Then one checks that F_{0}(z|τ) = F̄_{a}(e^{2πiz}|τ), F_{0}(z − 1|τ) = F̄_{f}(e^{2πiz}|τ) when (z, τ) ∈ {(z, τ)|τ ∈ iℝ_{+}^{×}, z = a + τb|a ∈ [0, 1], b ≥ 0}.

We then define F̄_{b}, ..., F̄_{e} as the solutions of (33) in D′ × iℝ_{+}^{×}, such that: F̄_{b}(u|τ) ≃ (1 − u)^{t} as u = 1 − 0^{+}, ℑ(u) > 0 for any τ, F̄_{c}(u|τ) ≃ uỹ as u → 0^{+}, ℑ(u) > 0 for any τ, F̄_{d}(u|τ) ≃ uỹ as u → 0^{+}, ℑ(u) < 0 for any τ, F̄_{e}(u|τ) ≃ (1 − u)^{t} as u = 1 − 0^{+}, ℑ(u) < 0 for any τ.

Then F̄_{b} = F̄_{a}(−2πi)^{t}, F̄_{c}(−|τ) = F̄_{b}(−|τ)[Φ(ỹ, t) + O(e^{2πiτ})], F̄_{d}(−|τ) = F̄_{c}(−|τ)e^{−2πiỹ}, F̄_{e}(−|τ) = F̄_{d}(−|τ)[Φ(ỹ, t)^{−1} + O(e^{2πiτ})], F̄_{f} = F̄_{e}(i/2π)^{t}.

So F̄_{f}(−|τ) = F̄_{a}(−|τ)((−2πi)^{t}Φ(ỹ, t)e^{−2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t} + O(e^{2πiτ})). It follows that F_{0}(z + 1|τ) = F_{0}(z|τ)A(τ), where

A(τ) = (−2πi)^{t}Φ(ỹ, t)e^{2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t} + O(e^{2πiτ}).

Now

Ã = F (z|τ)^{−1}F (z + 1|τ) = X(τ)^{−1}A(τ)X(τ)
= (1 + ∑_{k>0}(degree k)O(τ^{x(k)}e^{2πiτ}))^{−1}exp(\frac{τ}{2πi}([Δ] + a_{0}t))

((−2πi)^{t}Φ(ỹ, t)e^{2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t} + O(e^{2πiτ}))exp(−\frac{τ}{2πi}([Δ] + a_{0}t))

(1 + ∑_{k>0}(degree k)O(τ^{x(k)}e^{2πiτ})).

As we have seen, [Δ] + a_{0}t commutes with ỹ and t; on the other hand,

exp(\frac{τ}{2πi}([Δ] + a_{0}t))O(e^{2πiτ})exp(−\frac{τ}{2πi}([Δ] + a_{0}t))
= exp(τad(\frac{[Δ] + a_{0}t}{2πi}))(O(e^{2πiτ})) = ∑_{k≥0}(degree k)O(τ^{n_{1}(k))}e^{2πiτ})

where n_{1}(k) ≥ 0, as [Δ] + a_{0}t is a sum of terms of positive degree and of Δ_{0}, which is locally ad-nilpotent.

Then

Ã = (1 + ∑_{k>0}(degree k)O(τ^{x(k)}e^{2πiτ}))^{−1}((−2πi)^{t}Φ(ỹ, t)e^{2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t}

+ ∑_{k≥0}(degree k)O(τ^{n_{1}(k)}e^{2πiτ}))(1 + ∑_{k>0}(degree k)O(τ^{x(k)}e^{2πiτ})).

It follows that

Ã = (−2πi)^{t}Φ(ỹ, t)e^{2πiỹ}Φ(ỹ, t)^{−1}(i/2π)^{t} + ∑_{k≥0}(degree k)O(τ^{n_{2}(k)}e^{2πiτ}),

where n_{2}(k) ≥ 0, which implies the first formula for Ã. The second formula either follows from the first one by using the hexagon identity, or can be obtained repeating the above argument using a path 1 → +∞ → 1, winding around 1 and ∞. ⊓ ⊔

We now prove:

Theorem 26.

B̃ = (2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t}.

Proof. We first define F_{0}(z|τ) as the solution in V := {a + bτ|a ∈]0, 1[, b ∈ ℝ} of (30) such that F_{0}(z|τ) ∼ z^{t} as z → 0^{+}. Then there exists B(τ) such that e^{2πix}F_{0}(z + τ|τ) = F_{0}(z|τ)B(τ). We compute the asymptotics of B(τ) as τ → i∞.

We define four asymptotic zones (z is assumed to remain on the segment [0, τ], and τ in the line iℝ_{+}): (1) z ≪ 1 ≪ τ, (2) 1 ≪ z ≪ τ, (3) 1 ≪ τ −z ≪ τ,

(4) τ − z ≪ 1 ≪ τ.

In the transition (1)-(2), the system takes the form (32), or if we set u := e^{2πiz}, (33).

In the transition (3)-(4), G(z′|τ) := e^{2πix}F (τ + z′|τ) satisfies (30), so Ḡ(u′|τ) = e^{2πix}F (τ + z′|τ) satisfies (33), where u′ = e^{2πiz′}.

We now compute the form of the system in the transition (2)-(3). We first prove:

Lemma 27. Set u := e^{2πiz}, v := e^{2πi(τ−z)}. When 0 < ℑ(z) < ℑ(τ), we have |u| < 1, |v| < 1. When k ≥ 0, (θ^{(k)}/θ)(z|τ) = (−iπ)^{k} + ∑_{s,t≥0,s+t>0} a_{st}^{(k)}u^{s}v^{t}, where the sum in the r.h.s. is convergent in the domain |u| < 1, |v| < 1.

Proof. This is clear if k = 0. Set q = uv = e^{2πiτ}. We have θ(z|τ) = u^{1/2} ∏_{s>0}(1 − q^{s}u) ∏_{s≥0}(1 − q^{s}u^{−1}) · (2πi)^{−1} ∏_{s>0}(1 − q^{s})^{−2}, so

(θ′/θ)(z|τ) = iπ − 2πi ∑_{s>0} q^{s}u/(1 − q^{s}u) + 2πi ∑_{s≥0} q^{s}u^{−1}/(1 − q^{s}u^{−1})

= −iπ − 2πi ∑_{s≥0} \frac{u^{s+1}v^{s}}{1 − u^{s+1}v^{s}} + 2πi ∑_{s≥0} \frac{u^{s}v^{s+1}}{1 − u^{s}v^{s+1}} = −iπ + ∑_{s+t>0} a_{st}u^{s}v^{t}, where a_{st} = 2πi if (s, t) = k(r, r + 1), k > 0, r ≥ 0, and a_{st} = −2πi if (s, t) = k(r + 1, r), k > 0, r ≥ 0. One checks that this series is convergent in the domain |u| < 1, |v| < 1. This proves the lemma for k = 1.

We then prove the remaining cases by induction, using

\frac{θ^{(k+1)}}{θ} (z|τ) = \frac{θ^{(k)}}{θ} (z|τ)\frac{θ′}{θ} (z|τ) + \frac{∂}{∂z} \frac{θ^{(k)}}{θ} (z|τ).

⊓ ⊔

Using the expansion

\frac{θ(z + x|τ)x}{θ(z|τ)θ(x|τ)} = \frac{x}{θ(x|τ)} ∑_{k≥0}(θ^{(k)}/θ)(z|τ)\frac{x^{k}}{k!}

= \frac{πx}{sin(πx)}(1 + ∑_{n>0} q^{n}P_{n}(x))( ∑_{k≥0}((−iπ)^{k} + ∑_{s+t>0} a_{st}^{(k)}u^{s}v^{t})\frac{x^{k}}{k!} )

= \frac{πx}{sin(πx)}e^{−iπx} + ∑_{s+t>0} a_{st}(x)u^{s}v^{t} = \frac{2iπx}{e^{2iπx} − 1} + ∑_{s+t>0} a_{st}(x)u^{s}v^{t},

the form of the system in the transition (2)-(3) is

∂_{z}F (z|τ) = ( − \frac{2iπadx}{e^{2iπadx} − 1}(y) + ∑_{s,t|s+t>0} a_{st}u^{s}v^{t})F (z|τ)
= (2iπỹ + ∑_{s,t|s+t>0} a_{st}u^{s}v^{t})F(z|τ),  (34)

where each homogeneous part of ∑_{s,t} a_{st}u^{s}v^{t} converges for |u| < 1, |v| < 1.

Lemma 28. There exists a solution F_{c}(z|τ) of (34) defined for 0 < ℑ(z) < ℑ(τ), such that

F_{c}(z|τ) = uỹ(1 + ∑_{k>0} ∑_{s≤s(k)} log(u)^{s}f_{ks}(u, v))

(logu = iπz, uỹ = e^{2πizỹ}), where f_{ks}(u, v) is an analytic function taking its values in the homogeneous part of the algebra of degree k, convergent for |u| < 1 and |v| < 1, and vanishing at (0, 0). This function is uniquely defined up to right multiplication by an analytic function of the form 1 + ∑_{k>0} a_{k}(q) (recall that q = uv), where a_{k}(q) is an analytic function on {q||q| < 1}, vanishing at q = 0, with values in the degree k part of the algebra.

Proof of Lemma. We set G(z|τ) := u^{−ỹ}F (z|τ), so G(z|τ) should satisfy

∂_{z}G(z|τ) = exp(−ad(ỹ)logu){ ∑_{s+t>0} a_{st}u^{s}v^{t}}G(z|τ),

which has the general form

∂_{z}G(z|τ) = ( ∑_{k>0} ∑_{s≤a(k)} log(u)^{s}a_{ks}(u, v))G(z|τ),

where a_{ks}(u, v) is analytic in |u| < 1, |v| < 1 and vanishes at (0, 0). We show that this system admits a solution of the form 1+∑_{k>0} ∑_{s≤s(k)} log(u)^{s}f_{ks}(u, v), with f_{ks}(u, v) analytic in |u| < 1, |v| < 1, in the degree k part of the algebra, vanishing at (0, 0) for s ≠ 0. For this, we solve inductively (in k) the system of equations

∂_{z}( ∑_{s}(logu)^{s}f_{ks}(u, v)) = ∑_{s′,s′′,k′,k′′|k′+k′′=k}(logu)^{s′+s′′}a_{k′s′}(u, v)f_{k′′s′′}(u, v).

(35)

Let O be the ring of analytic functions on {(u, v)||u| < 1, |v| < 1} (with values in a finite dimensional vector space) and 𝔪 ⊂ O be the subset of functions vanishing at (0, 0). We have an injection O[X] → {analytic functions in (u, v), |u| < 1, |v| < 1, u ∉ ℝ_{−}}, given by f(u, v)X^{k} ↦ (logu)^{k}f(u, v). The endomorphism \frac{∂}{∂z} = 2πi(u\frac{∂}{∂u} −v \frac{∂}{∂v} ) then corresponds to the endomorphism of O[X] given by 2πi(\frac{∂}{∂X} +u\frac{∂}{∂u} −v \frac{∂}{∂v} ). It is surjective, and restricts to a surjective endomorphism of 𝔪[X]. The latter surjectivity implies that equation (35) can be solved.

Let us show that the solution G(z|τ) is unique up to right multiplication by functions of q like in the lemma. The ratio of two solutions is of the form 1 + ∑_{k>0} ∑_{s≤s(k)} log(u)^{s}f_{ks}(u, v) and is killed by ∂_{z}. Now the kernel of the endomorphism of 𝔪[X] given by 2πi(\frac{∂}{∂X} + u\frac{∂}{∂u} − v \frac{∂}{∂v} ) is m^{∗}(𝔪_{1}), where m^{∗}(𝔪_{1}) ⊂ 𝔪 is the set of all functions of the form a(uv), where a is an analytic function on {q||q| < 1} vanishing at 0. This implies that the ratio of two solutions is as above. ⊓ ⊔

End of proof of Theorem. Similarly, there exists a solution F_{d}(z|τ) of (34) defined in the same domain, such that

F_{d}(z|τ) = v^{−ỹ}(1 + ∑_{k>0} ∑_{s≤t(k)} log(v)^{t}g_{ks}(u, v)),

where b_{ks}(u, v) is as above (and logv = iπ(τ − z), v^{−ỹ} = exp(2πi(z − τ)ỹ)). F_{d}(z|τ) is defined up to right multiplication by a function of q as above.

We now study the ratio F_{c}(z|τ)^{−1}F_{d}(z|τ). This is a function of τ only, and it has the form

q^{−ỹ}(1 + ∑_{k>0} ∑_{s≤s(k),t≤t(k)}(logu)^{s}(logv)^{t}a_{kst}(u, v))

where a_{kst}(u, v) ∈ 𝔪 (as v^{−ỹ}(1+∑_{k>0} ∑_{s≤s(k)}(logu)^{s}c_{ks}(u, v))vỹ has the form 1 + ∑_{k>0} ∑_{s,t≤t(k)} (logu)^{s}(logv)^{t}d_{ks}(u, v), where d_{ks}(u, v) ∈ 𝔪 if c_{ks}(u, v) ∈ 𝔪). Set logq := logu + logv = 2πiτ, then this ratio can be rewritten q^{−ỹ}{1 + ∑_{k>0} ∑_{s≤s(k),t≤t(k)}(logu)^{s}(logq)^{t}b_{kst}(u, v)} where b_{kst}(u, v) ∈ 𝔪, and since the product of this ratio with qỹ is killed by ∂_{z} (which identifies with the endomorphism 2πi(\frac{∂}{∂X} + u\frac{∂}{∂u} − v \frac{∂}{∂v} ) of O[X]), the ratio is in fact of the form

F_{c}^{−1}F_{d}(z|τ) = q^{ỹ}(1 + ∑_{k>0} ∑_{s≤s(k)}(logq)^{s}a_{ks}(q)),

where a_{ks} is analytic in {q||q| < 1}, vanishing at q = 0.

It follows that

F_{c}^{−1}F_{d}(z|τ) = e^{−2πiτỹ}(1 + ∑_{k>0}(degree k)O(τ^{k}e^{−2πiτ})).  (36)

In addition to F_{c} and F_{d}, which have prescribed behaviors in zones (2) and (3), we define solutions of (30) in V by prescribing behaviors in the remaining asymptotic zones: F_{a}(z|τ) ≃ z^{t} when z → 0^{+} for any τ; F_{b}(z|τ) ≃ (2πz/i)^{t} when z → i0^{+} for any τ (in particular in zone (1)); e^{2πix}F_{e}(z|τ) ≃ (2π(τ − z)/i)^{t} when z = τ − i0^{+} for any τ; e^{2πix}F_{f}(z|τ) ≃ (z − τ)^{t} when z = τ + 0^{+} for any τ (in particular in zone (4)).

Then F_{0}(z|τ) = F_{a}(z|τ), and e^{−2πix}F_{0}(z − τ|τ) = F_{f}(z|τ). We have F_{b} = F_{a}(2π/i)^{t}, F_{f} = F_{e}(2πi)^{−t}.

Let us now compute the ratio between F_{b} and F_{c}. Recall that u = e^{2πiz}, v = e^{2πi(τ−z)}. Set F̄(u, v) := F (z|τ). Using the expansion of θ(z|τ), one shows that (30) has the form

∂_{u}F̄(u, v) = (\frac{A(u, v)}{u} + \frac{B(u, v)}{u − 1})F̄(u, v),

where A(u, v) is holomorphic in the region |v| < 1/2, |u| < 2, and A(u, 0) = ỹ, B(u, 0) = t. We have F̄_{b}(u, v) = (1 − u)^{t}(1 + ∑_{k} ∑_{s≤s(k)} log(1 − u)^{k}b_{ks}(u, v)) and F̄_{b}(u, v) = ut̃(1+∑_{k} ∑_{s≤s(k)} log(u)^{k}a_{ks}(u, v)), with a_{ks}, b_{ks} analytic, and a_{ks}(0, v) = b_{ks}(1, v) = 0. The ratio F̄_{b}^{−1}F̄_{c} is an analytic function of q only, which coincides with Φ(ỹ, t) for q = 0, so it has the form Φ(ỹ, t)+∑_{k>0} a_{k}(q), where a_{k}(q) has degree k, is analytic in the neighborhood of q = 0 and vanishes at q = 0. Therefore

F_{c}(z|τ) = F_{b}(z|τ)(Φ(ỹ, t) + O(e^{2πiτ})).

In the same way, one proves that

F_{e}(z|τ) = F_{d}(e^{−2πix}Φ(−ỹ − t, t)^{−1} + O(e^{2πiτ})).

Indeed, let us set Ḡ_{d}(u′, v′) := e^{2πix}F_{d}(τ + z′|τ), Ḡ_{e}(u′, v′) := e^{2πix}F_{e}(τ + z′|τ), where u′ = e^{2πi(τ+z′)}, v′ = e^{−2πiz′}, then Ḡ_{d}(u′, v′) ≃ (v′)^{−ỹ−t}e^{2πix} as (u′, v′) → (0^{+}, 0^{+}) and Ḡ_{e}(u′, v′) ≃ (1 − v′)^{t} as v′ → 1^{−} for any u′, and both Ḡ_{d} and Ḡ_{e} are solutions of ∂_{v′} Ḡ(u′, v′) = [−(ỹ+t)/v′+t/(v′−1)+O(u′)]Ḡ(v′). Therefore Ḡ_{d} = Ḡ_{e}[Φ(−ỹ − t, t)e^{2πix} + O(u′)].

Combining these results, we get:

Lemma 29.

B(τ) ≃ (2πi)^{t}Φ(−ỹ − t, t)e^{2πix}e^{2iπτỹ}Φ(ỹ, t)^{−1}(2π/i)^{−t},

in the sense that the left (equivalently, right) ratio of these quantities has the form 1 + ∑_{k>0}(degree k)O(τ^{n(k)}e^{2πiτ}) for n(k) ≥ 0.

Recall that we have proved:

F (z|τ) = F_{0}(z|τ) exp(−\frac{a_{0}}{2πi}(∫_{i}^{τ} E_{2} + C)t)[F (τ)],

where C is such that ∫_{i}^{τ} E_{2} + C = τ + O(e^{2πiτ}).

Set X(τ) := exp(−\frac{a_{0}}{2πi}(∫_{i}^{τ} E_{2} + C)t)[F (τ)]. When τ → i∞, X(τ) = exp(−\frac{τ}{2πi}([Δ] + a_{0}t))(1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ})). Then

B̃ = F (z|τ)^{−1}e^{2πix}F (z + τ|τ) = X(τ)^{−1}B(τ)X(τ)

= Ad((1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}))^{−1}exp(\frac{τ}{2πi}([Δ] + a_{0}t)))

(((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}e^{2πiτỹ}Φ(ỹ, t)^{−1}(2π/i)^{−t})(1 + ∑_{k>0}(degree k)O(τ^{n(k)}e^{2πiτ}))),

where Ad(u)(x) = uxu^{−1}.

[Δ] + a_{0}t commutes with ỹ and t; assume for a moment that

Ad(exp(\frac{τ}{2πi}([Δ] + a_{0}t)))(e^{2πix}e^{2πiτỹ}) = e^{2πix}

(Lemma 30 below), then

Ad(exp(\frac{τ}{2πi}([Δ] + a_{0}t)))((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}e^{2πiτỹ}Φ(ỹ, t)^{−1}(2π/i)^{−t})

= (2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t}.

On the other hand, Ad(exp(\frac{τ}{2πi}([Δ]+a_{0}t)))(1+∑_{k>0}(degree k)O(τ^{n(k)}e^{2πiτ})) has the form 1 + ∑_{k>0}(degree k)O(τ^{n′(k)}e^{2πiτ}), where n′(k) ≥ 0. It follows that

B̃ = Ad(1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}))

(((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t})(1 + ∑_{k>0}(degree k)O(τ^{n′(k)}e^{2πiτ})));

now

Ad((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t})^{−1}(1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}))

= 1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}), so

B̃ = ((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t})(1 + ∑_{k>0}(degree k)O(τ^{f(k)}e^{2πiτ}))
(1 + ∑_{k>0}(degree k)O(τ^{n′(k)}e^{2πiτ}))
= ((2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t})(1 + ∑_{k>0}(degree k)O(τ^{n′′(k)}e^{2πiτ}))

for n′′(k) ≥ 0. Since B̃ is constant w.r.t. τ, this implies

B̃ = (2πi)^{t}Φ(−ỹ − t, t)e^{2πix}Φ(ỹ, t)^{−1}(2π/i)^{−t},

as claimed.

We now prove the conjugation used above.

Lemma 30. For any τ ∈ ℂ, we have

e^{\frac{τ}{2πi} ([Δ]+a_{0}t)}e^{2πix}e^{− \frac{τ}{2πi} ([Δ]+a_{0}t)}e^{2iπτỹ} = e^{2πix}.

Proof. We have [Δ] + a_{0}t = Δ_{0} + ∑_{k≥0} a_{2k}(δ_{2k} + (adx)^{2k}(t)) (where δ_{0} = 0), so [[Δ] + a_{0}t, x] = y − ∑_{k≥0} a_{2k}(adx)^{2k+1}(t). Recall that

∑_{k≥0} a_{2k}u^{2k} = \frac{π^{2}}{sin^{2}(πu)} − \frac{1}{u^{2}} ,

then [[Δ] + a_{0}t, x] = y − (adx)(\frac{π^{2}}{sin^{2}(πadx)} − \frac{1}{(adx)^{2}} )(t). So

e^{−2πix}(\frac{1}{2πi}([Δ] + a_{0}t))e^{2πix}

= \frac{1}{2πi}([Δ] + a_{0}t) + \frac{e^{−2πiadx} − 1}{adx}([x, \frac{1}{2πi}([Δ] + a_{0}t)])

= \frac{1}{2πi}([Δ] + a_{0}t) − \frac{1}{2πi} \frac{e^{−2πiadx} − 1}{adx}(y − (adx)(\frac{π^{2}}{sin^{2}(πadx)} − \frac{1}{(adx)^{2}} )(t)).

We have

−\frac{1}{2πi} \frac{e^{−2πiadx} − 1}{adx}(y − (adx)(\frac{π^{2}}{sin^{2}(πadx)} − \frac{1}{(adx)^{2}} )(t)) = −2πiỹ,

therefore we get

e^{−2πix}(\frac{1}{2πi}([Δ] + a_{0}t))e^{2πix} = \frac{1}{2πi}([Δ] + a_{0}t) − 2πiỹ.

Multiplying by τ, taking the exponential, and using the fact that [Δ] + a_{0}t commutes with ỹ, we get

e^{−2πix}e^{\frac{τ}{2πi} ([Δ]+a_{0}t)}e^{2πix} = e^{\frac{τ}{2πi} ([Δ]+a_{0}t)}e^{−2πiτỹ},

which proves the lemma. ⊓ ⊔

This ends the proof of Theorem 26.

5 Construction of morphisms Γ_{1,[n]} → G_{n} ⋊ S_{n}

In this section, we fix a field k of characteristic zero. We denote the algebras 𝔱̄_{1,n}^{k}, 𝔱_{n}^{k} simply by 𝔱̄_{1,n}, 𝔱_{n}. The above group G_{n} is the set of ℂ-points of a group scheme defined over ℚ, and we now again denote by G_{n} the set of its k-points.

5.1 Construction of morphisms Γ_{1,[n]} → G_{n} ⋊ S_{n} from a 5-uple (Φ_{λ}, Ã, B̃, Θ̃, Ψ̃)

Let Φ_{λ} be a λ-associator defined over k. This means that Φ_{λ} ∈ exp(𝔱̂_{3}) (the Lie algebras are now over k),

Φ_{λ}^{3,2,1} = Φ_{λ}^{−1}, Φ_{λ}^{2,3,4}Φ_{λ}^{1,23,4}Φ_{λ}^{1,2,3} = Φ_{λ}^{1,2,34}Φ_{λ}^{12,3,4},  (37)
e^{λt_{31}/2}Φ_{λ}^{2,3,1}e^{λt_{23}/2}Φ_{λ}e^{λt_{12}/2}Φ_{λ}^{3,1,2} = e^{λ(t_{12}+t_{23}+t_{13})/2}.  (38)

E.g., the KZ associator is a 2πi-associator over ℂ.

Proposition 31. If Θ̃, Ψ̃ ∈ G_{1} and Ã, B̃ ∈ exp(ˆ𝔱̄_{1,2}) satisfy: the “Γ_{1,1} identities” (27), the “Γ_{1,2} identities” (28), (29), and the “Γ_{1,[3]} identities” (23), (22), (26) (with 2πi replaced by λ), as well as Ã^{∅,1} = Ã^{1,∅} = B̃^{∅,1} = B̃^{1,∅} = 1, then one defines a morphism Γ_{1,[n]} → G_{n} ⋊ S_{n} by

Θ ↦ [Θ̃]e^{i \frac{π}{2} ∑_{i<j} t̄_{ij}}, Ψ ↦ [Ψ̃]e^{i \frac{π}{6} ∑_{i<j} t̄_{ij}},
σ_{i} ↦ {Φ_{λ}^{1...i−1,i,i+1}}^{−1}e^{λt̄_{i,i+1}/2}(i, i + 1){Φ_{λ}^{1...i−1,i,i+1}},

C_{jk} ↦ {Φ_{λ,j}^{−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(e^{λt_{12}})^{j...k−1,k...n}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j}},

A_{i} ↦ {Φ_{λ,i}}^{−1}Ã^{1...i−1,i...n}{Φ_{λ,i}}, B_{i} ↦ {Φ_{λ,i}}^{−1}B̃^{1...i−1,i...n}{Φ_{λ,i}},

where Φ_{λ,i} = Φ_{λ}^{1...i−1,i,i+1...n}...Φ_{λ}^{1...n−2,n−1,n}.

According to Section 4.4, the representations γ_{n} are obtained by the procedure described in this proposition from the KZ associator, Θ̃, Ψ̃ arising from γ_{1}, and Ã, B̃ arising from γ_{2}.

Note also that the analogue of (22) is equivalent to the pair of equations

e^{λt̄_{12}/2}Ã^{2,1}e^{λt̄_{12}/2}Ã = 1,
(e^{λt̄_{12}/2}Ã)^{3,12}Φ_{λ}^{3,1,2}(e^{λt̄_{12}/2}Ã)^{2,31}Φ_{λ}^{2,3,1}(e^{λt̄_{12}/2}Ã)^{1,23}Φ_{λ}^{1,2,3} = 1,

and similarly (23) is equivalent to the same equations, with Ã, λ replaced by B̃, −λ.

Remark 32. One can prove that it Φ_{λ} satisfies only the pentagon equation and Θ̃, Ψ̃, Ã, B̃ satisfy the the “Γ_{1,1} identities” (27), the “Γ_{1,2} identities” (28), (29), and the “Γ_{1,3} identities” (24), (26), then the above formulas (removing σ_{i}) define a morphism Γ_{1,n} → G_{n}. In the same way, if Φ_{λ} satisfies all the associator conditions and Ã, B̃ satisfy the Γ_{1,[3]} identities (22), (23), (26), then the above formulas (removing Θ, Ψ) define a morphism \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n})⋊S_{n}.

Proof. Let us prove that the identity (A_{i}, A_{j}) = 1 (i < j) is preserved. Applying x ↦ x^{1...i−1,i...j−1,j...n} to the first identity of (24), we get

(Ã^{1...i−1,i...n}, Φ_{λ}^{1...,i...j−1,...n}Ã^{1...j−1,j...n}(Φ_{λ}^{−1})^{1...,i...j−1,...n}) = 1.

The pentagon identity implies

Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n}  (39)
= (Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,..,n})Φ_{λ}^{1...,i...j−1,...n}(Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,j−1}),

so the above identity is rewritten

(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,..,n}Ã^{1...i−1,i...n}(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,..,n})^{−1},
Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n}(Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,...j−1})^{−1}Ã^{1...j−1,j...n}
Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,...j−1}(Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n})^{−1}) = 1.

Now Φ_{λ}^{i,i+1,...n}, …, Φ_{λ}^{i...,j−1,..,n} commute with Ã^{1...i−1,i...n}, and Φ_{λ}^{1...,i,...j−1}, …, Φ_{λ}^{1...,j−2,...j−1} commute with Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,...j−1}, which implies

(Ã^{1...i−1,i...n}, Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n}Ã^{1...j−1,j...n}(Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n})^{−1}) = 1,

so that (A_{i}, A_{j}) = 1 is preserved. In the same way, one shows that (B_{i}, B_{j}) = 1 is preserved.

Let us show that (B_{k}, A_{k}A_{j}^{−1}) = C_{jk} is preserved (if j ≤ k).

(Φ_{λ,k}^{−1}B̃^{1...k−1,k...n}Φ_{λ,k}, Φ_{λ,k}^{−1}Ã^{1...k−1,k...n}Φ_{λ,k}Φ_{λ,j}^{−1}(Ã^{1...j−1,j...n})^{−1}Φ_{λ,j})

= Φ_{λ,j}^{−1}((Φ_{λ}^{1...,j,...n}...Φ_{λ}^{1...,k−1,...n})B̃^{1...k−1,k...n}(Φ_{λ}^{1...,j,...n}...Φ_{λ}^{1...,k−1,...n})^{−1},

(Φ_{λ}^{1...,j,...n}...Φ_{λ}^{1...,k−1,...n})Ã^{1...k−1,k...n}(Φ_{λ}^{1...,j,...n}...Φ_{λ}^{1...,k−1,...n})^{−1}(Ã^{1...j−1,j...n})^{−1})Φ_{λ,j}

= Φ_{λ,j}^{−1}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,j...k−1,...n}B̃^{1...k−1,k...n}

(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,j...k−1,...n})^{−1}, Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,j...k−1,...n} Ã^{1...k−1,k...n}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,j...k−1,...n})^{−1}(Ã^{1...j−1,j...n})^{−1})Φ_{λ,j}

= Φ_{λ,j}^{−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(Φ_{λ}^{1...,j...k−1,...n}B̃^{1...k−1,k...n}(Φ_{λ}^{1...,j...k−1,...n})^{−1},

Φ_{λ}^{1...,j...k−1,...n}Ã^{1...k−1,k...n}(Φ_{λ}^{1...,j...k−1,...n})^{−1}(Ã^{1...j−1,j...n})^{−1})

(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j}

= Φ_{λ,j}^{−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}{Φ(B̃^{12,3}, Ã^{12,3}Φ_{λ}^{−1}(Ã^{1,23})^{−1}Φ_{λ})Φ_{λ}^{−1}}^{1...,j...k−1,...n}

(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j}

= Φ_{λ,j}^{−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(e^{2πit̄_{12}})^{j...k−1,k...n}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j},

where the second identity uses (39) and the invariance of Φ_{λ}, the third identity uses the fact that Φ_{λ}^{j,j+1,...n}, ..., Φ_{λ}^{j...,k−1,...n} commute with Ã^{1...j−1,j..n} (again by the invariance of Φ_{λ}), and the last identity uses (26). So (B_{k}, A_{k}A_{j}^{−1}) = C_{jk} is preserved. One shows similarly that

(Φ_{λ,k}^{−1}B̃^{1...k−1,k...n}Φ_{λ,k}Φ_{λ,j}^{−1}(B̃^{1...j−1,j...n})^{−1}Φ_{λ,j}, Φ_{λ,k}^{−1}Ã^{1...k−1,k...n}Φ_{λ,k})

= Φ_{j}^{−1}Φ^{j,j+1,...n}...Φ^{j...,k−1,...n}(e^{2πit̄_{12}})^{j...k−1,k...n}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j},

so that (B_{k}B_{j}^{−1}, A_{k}) = C_{jk} is preserved.

Let us show that (A_{i}, C_{jk}) = 1 (i ≤ j ≤ k) is preserved. We have

(Φ_{λ,i}^{−1}Ã^{1...i−1,i...n}Φ_{λ,i}, Φ_{λ,j}^{−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(e^{2πit̄_{12}})^{j...k−1,k...n}

(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1}Φ_{λ,j})

= Φ_{λ,i}^{−1}(Ã^{1...i−1,i...n}, Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(e^{2πit̄_{12}})^{j...k−1,k...n}

(Φ_{λ}^{1...,i,...n}...Φ_{λ}^{1...,j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1})Φ_{λ,i}

= Φ_{λ,i}^{−1}(Ã^{1...i−1,i...n}, Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,j−1}

Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}(e^{2πit̄_{12}})^{j...k−1,k...n}

(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{1...,i,...j−1}...Φ_{λ}^{1...,j−2,j−1}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1})Φ_{λ,i}

= Φ_{λ,i}^{−1}(Ã^{1...i−1,i...n}, Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}

(e^{2πit̄_{12}})^{j...k−1,k...n}(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1})Φ_{λ,i}

= Φ_{λ,i}^{−1}Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}(Ã^{1...i−1,i...n}, Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}

(e^{2πit̄_{12}})^{j...k−1,k...n}(Φ_{λ}^{1...,i...j−1,...n}Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n})^{−1})(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n})^{−1}Φ_{λ,i}

= Φ_{λ,i}^{−1}Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n}(Ã^{1...i−1,i...n}, Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,i...j−1,...n}

(e^{2πit̄_{12}})^{j...k−1,k...n}(Φ_{λ}^{j,j+1,...n}...Φ_{λ}^{j...,k−1,...n}Φ_{λ}^{1...,i...j−1,...n})^{−1})(Φ_{λ}^{i,i+1,...n}...Φ_{λ}^{i...,j−1,...n})^{−1}Φ_{λ,i}

= 1,

where the second equality follows from the generalized pentagon identity (39), the third equality follows from the fact that Φ_{λ}^{1...,i,...j−1}, ..., Φ_{λ}^{1...,j−2,j−1} commute with (e^{2πit̄_{12}})^{j...k−1,k...n}, Φ_{λ}^{j,j+1,...n}, ..., Φ_{λ}^{j...,k−1,...n}, the fourth equality follows from the fact that Φ_{λ}^{i,i+1,...n}, ..., Φ_{λ}^{i...,j−1,...n} commute with Ã^{1...i−1,i...n} (as Φ_{λ} is invariant), the last equality follows from the fact that Φ_{λ}^{1...,i...j−1,j...n} commutes with Φ_{λ}^{j,j+1,...n}, ..., Φ_{λ}^{j...,k−1,...n} (again as Φ_{λ} is invariant) and with (e^{2πit̄_{12}})^{j...k−1,k...n} (as t_{34} commutes with the image of 𝔱_{3} → 𝔱_{4}, x ↦ x^{1,2,34}). Therefore (A_{i}, C_{jk}) = 1 is preserved. One shows similarly that (B_{i}, C_{jk}) = 1 (i ≤ j ≤ k), X_{i+1} = σ_{i}X_{i}σ_{i} and Y_{i+1} = σ_{i}^{−1}Y_{i}σ_{i}^{−1} are preserved.

The fact that the relations ΘA_{i}Θ^{−1} = B_{i}^{−1}, ΘB_{i}Θ^{−1} = B_{i}A_{i}B_{i}^{−1}, ΨA_{i}Ψ^{−1} = A_{i}, ΨB_{i}Ψ^{−1} = B_{i}A_{i}, are preserved follows from the identities (28), (29) and that if we denote by x ↦ [x]_{n} the morphism 𝔡 → 𝔡 ⋊ 𝔱̄_{1,n} defined above, then: (a) Φ_{i} commutes with ∑_{i,j|i<j} t̄_{ij} and with the image of 𝔡 → 𝔡 ⋊ 𝔱̄_{1,n}, x ↦ [x]_{n}; (b) for x ∈ 𝔡, y ∈ 𝔱̄_{1,2}, we have [[x]_{n}, y^{1...i−1,i...n}] = [[x]_{2}, y]^{1...i−1,i...n}. Let us prove (a): the first part follows from the fact that Φ commutes with t_{12} + t_{13} + t_{23}; the second part follows from the fact that X, d, Δ_{0} and δ_{2n} + ∑_{k<l}(adx̄_{k})^{2n}(t̄_{kl}) commute with t̄_{ij} for any i < j. Let us prove (b): the identity holds for [x, x′] whenever it holds for x and for x′, so it suffices to check it for x a generator of 𝔡; x being such a generator, both sides are (as functions of y) derivations 𝔱̄_{1,2} → 𝔱̄_{1,n} w.r.t. the morphism 𝔱̄_{1,2} → 𝔱̄_{1,n}, y ↦ y^{1...i−1,i...n}, so it suffices to check the identity for y a generator of 𝔱̄_{1,2}. The identity is obvious if x ∈ {Δ_{0}, d, X} and y ∈ {x̄_{1}, ȳ_{1}, x̄_{2}, ȳ_{2}}. If x = δ_{2s} and y = x̄_{1}, then the identity holds because we have

[δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), x̄_{1}]^{1...i−1,i...n} = −((adx̄_{1})^{2s+1}(t̄_{12}))^{1...i−1,i...n}
= −(ad( ∑_{u′=1}^{i−1} x̄_{u′}))^{2s+1}( ∑_{1≤u<i≤v≤n} t̄_{uv})
= − ∑_{1≤u<i≤v≤n}(adx̄_{u})^{2s+1}(t̄_{uv}),

while

[δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ∑_{u′=1}^{i−1} x̄_{u′}] = [ ∑_{1≤u<i≤v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ∑_{u′=1}^{i−1} x̄_{u′}]
= − ∑_{1≤u<i≤v≤n}(adx̄_{u})^{2s+1}(t̄_{uv})

where the first equality follows from the fact that (adx̄_{u})^{2s}(t̄_{uv}) commutes with ∑_{u′=1}^{i−1} x̄_{u′} whenever u < v < i or i ≤ u < v. If x = δ_{2s} and y = x̄_{2}, then the identity follows because [δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), x̄_{1} + x̄_{2}] = 0 and [δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ∑_{u′=1}^{n} x̄_{u′}] = 0.

If x = δ_{2s} and y = ȳ_{1}, then

[δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), ȳ_{1}]^{1...i−1,i...n}

= \{\frac{1}{2} ∑_{p+q=2s−1}[(adx̄_{1})^{p}(t̄_{12}), (−adx̄_{1})^{q}(t̄_{12})] + [(adx̄_{1})^{2s}(t̄_{12}), ȳ_{1}]\}^{1...i−1,i...n}

= \frac{1}{2} ∑_{p+q=2s−1}[ ∑_{1≤u<i≤v≤n}(adx̄_{u})^{p}(t̄_{uv}), ∑_{1≤u′<i≤v′≤n}(adx̄_{u′})^{q}(t̄_{u′v′})]

+[ ∑_{1≤u<i≤v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}];

on the other hand,

[δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]
= ∑_{1≤u<v≤n}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

+ ∑_{u=1}^{i−1} ∑_{v|v≠u} ∑_{p+q=2s−1} \frac{1}{2}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u})^{q}(t̄_{uv})]
= ∑_{1≤u<v≤n}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

+ ∑_{1≤u<i≤v≤n} ∑_{p+q=2s−1} \frac{1}{2}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u})^{q}(t̄_{uv})],

where the second equality follows from the fact that [(adx̄_{u})^{p}(t̄_{uv}), (−x̄_{u})^{q}(t̄_{uv})]+ [(adx̄_{v})^{p}(t̄_{uv}), (−adx̄_{v})^{q}(t̄_{uv})] = 0 as p + q is odd.

Then

[δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), ȳ_{1}]^{1...i−1,i...n} − [δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

= − ∑_{1≤u<v<i}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}] − ∑_{i≤u<v≤n}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

+\frac{1}{2} ∑_{p+q=2s−1} ∑_{\begin{matrix}1≤u<i≤v≤n \\ 1≤u′<i≤v′≤n,(u,v)≠(u′,v′) \end{matrix}}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u′})^{q}(t̄_{u′v′})]

= ∑_{1≤u<v<i}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{i} + ... + ȳ_{n}] − ∑_{i≤u<v≤n}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

+\frac{1}{2} ∑_{p+q=2s−1} ∑_{\begin{matrix}1≤u<i≤v≤n \\ 1≤u<i≤v′≤n,v≠v′ \end{matrix}}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u})^{q}(t̄_{uv′})]

+\frac{1}{2} ∑_{p+q=2s−1} ∑_{\begin{matrix}1≤u<i≤v≤n \\ 1≤u′<i≤v≤n,u≠u′ \end{matrix}}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u′})^{q}(t̄_{u′v})]

where the second equality follows from the centrality of ȳ_{1} + ... + ȳ_{n}, the last equality follows for the fact that (adx̄_{u})^{p}(t̄_{uv}) and (−adx̄_{u′})^{q}(t̄_{u′v′}) commute for u, v, u′, v′ all distinct. Since p + q is odd, it follows that [δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), ȳ_{1}]^{1...i−1,i...n} − [δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

= ∑_{1≤u<v<i}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{i} + ... + ȳ_{n}] − ∑_{i≤u<v≤n}[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}]

+ ∑_{p+q=2s−1} ∑_{1≤u<i≤v<v′≤n}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u})^{q}(t̄_{uv′})]

+ ∑_{p+q=2s−1} ∑_{1≤u<u′<i≤v≤n}[(adx̄_{u})^{p}(t̄_{uv}), (−adx̄_{u′})^{q}(t̄_{u′v})].

Now if 1 ≤ u < v < i, we have

[(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{i} + ... + ȳ_{n}] = ∑_{p+q=2s−1}(adx̄_{u})^{p}ad(t̄_{ui} + ... + t̄_{un})(adx̄_{u})^{q}(t̄_{uv})

= ∑_{w=i}^{n} ∑_{p+q=2s−1}(adx̄_{u})^{p}[t̄_{uw}, (−adx̄_{v})^{q}(t̄_{uv})]

= ∑_{w=i}^{n} ∑_{p+q=2s−1}(adx̄_{u})^{p}(−adx̄_{v})^{q}([t̄_{uw}, t̄_{uv}])

= − ∑_{w=i}^{n} ∑_{p+q=2s−1}(adx̄_{u})^{p}(−adx̄_{v})^{q}([t̄_{uw}, t̄_{vw}])

= − ∑_{w=i}^{n} ∑_{p+q=2s−1}[(adx̄_{u})^{p}(t̄_{uw}), (−adx̄_{v})^{q}(t̄_{vw})];

one shows in the same way that if i ≤ u < v ≤ n, then [(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{i−1}] = ∑_{w=1}^{i−1} ∑_{p+q=2s−1}[(adx̄_{u})^{p}(t̄_{uw}), (−adx̄_{v})^{q}(t̄_{vw})]; all this implies that

[δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), ȳ_{1}]^{1...i−1,i...n} − [δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), (ȳ_{1})^{1...i−1}].

Since [δ_{2s} +(adx̄_{1})^{2s}(t̄_{12}), ȳ_{1} +ȳ_{2}] = 0 and [δ_{2s} +∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), ȳ_{1} + ... + ȳ_{n}] = 0, this equality implies

[δ_{2s} + (adx̄_{1})^{2s}(t̄_{12}), ȳ_{2}]^{1...i−1,i...n} − [δ_{2s} + ∑_{1≤u<v≤n}(adx̄_{u})^{2s}(t̄_{uv}), (ȳ_{2})^{1...i−1}],

which ends the proof of (b) above, and therefore of the fact that the identities ΘA_{i}Θ^{−1} = B_{i}^{−1}, ..., ΨB_{i}Ψ^{−1} = B_{i}A_{i} are preserved.

The relation (Θ, Ψ^{2}) = 1 is preserved because

([Θ̃]e^{i \frac{π}{2} ∑_{i<j} t̄_{ij}}, ([Ψ̃]e^{i \frac{π}{6} ∑_{i<j} t̄_{ij}})^{2}) = ([Θ̃]e^{i \frac{π}{2} ∑_{i<j} t̄_{ij}}, [Ψ̃]^{2}e^{i \frac{π}{3} ∑_{i<j} t̄_{ij}})
= ([Θ̃], [Ψ̃]^{2}) = [(Θ̃, Ψ̃^{2})] = 1,

where the two first identities follow from the fact that ∑_{i<j} t̄_{ij} commutes with the image of 𝔡 → 𝔡 ⋊ 𝔱̄_{1,n}, x ↦ [x], the third identity follows from the fact that G_{1} → G_{n}, g ↦ [g] is a group morphism, and the last identity follows from (27).

The image of C_{i,i+1} is Φ_{λ,i}^{−1}(e^{2πit̄_{12}})^{i,i+1...n}Φ_{λ,i}, to the product of the images of C_{12}, ..., C_{n−1,n} is

Φ_{λ,1}^{−1}(e^{2πit̄_{12}})^{1,2...n}(Φ_{λ,1}Φ_{λ,2}^{−1})(e^{2πit̄_{12}})^{2,3...n}(Φ_{λ,2}Φ_{λ,3}^{−1})(e^{2πit̄_{12}})^{3,4...n}...(Φ_{λ,n−1}Φ_{λ,n}^{−1})

e^{2πit̄_{n−1,n}}Φ_{λ,n}

= Φ_{λ,1}^{−1}(e^{2πit̄_{12}})^{1,2...n}(e^{2πit̄_{12}})^{2,3...n}Φ_{λ}^{1,2,3...n}(e^{2πit̄_{12}})^{3,4...n}...Φ_{λ}^{1...,i−1,...n}

(e^{2πit̄_{12}})^{i,i+1...n}...Φ_{λ}^{1...,n−2,n−1 n}e^{2πit̄_{n−1,n}}

= Φ_{λ,1}^{−1}(e^{2πit̄_{12}})^{1,2...n}(e^{2πit̄_{12}})^{2,3...n}(e^{2πit̄_{12}})^{3,4...n}...(e^{2πit̄_{12}})^{i,i+1...n}...e^{2πit̄_{n−1,n}}

Φ_{λ}^{1,2,3...n}...Φ_{λ}^{1...,i−1,...n}...Φ_{λ}^{1...,n−2,n−1 n}

= Φ_{λ,1}^{−1}e^{2πi ∑_{i<j} t̄_{ij}}Φ_{λ,1} = e^{2πi ∑_{i<j} t̄_{ij}},

where the second equality follows from the fact that Φ^{1...,i,...n} commutes with (e^{2πit̄_{12}})^{j,j+1...n} whenever j > i, and the last equality follows from the fact that ∑_{i<j} t_{ij} is central is 𝔱_{n}.

So the product of the images of C_{12}...C_{n−1,n} is e^{2πi ∑_{i<j} t̄_{ij}}.

The relation (ΘΨ)^{3} = C_{12}...C_{n−1,n} is then preserved because

([Θ̃]e^{i \frac{π}{2} ∑_{i<j} t̄_{ij}}[Ψ̃]e^{i \frac{π}{6} ∑_{i<j} t̄_{ij}})^{3} = ([Θ̃][Ψ̃])^{3}e^{2πi ∑_{i<j} t̄_{ij}} = [(Θ̃Ψ̃)^{3}]e^{2πi ∑_{i<j} t̄_{ij}}
= e^{2πi ∑_{i<j} t̄_{ij}},

where the first equality follows from the fact that ∑_{i<j} t̄_{ij} commutes with the image of G_{1} → G_{n}, g ↦ [g], the second equality follows from the fact that g ↦ [g] is a group morphism and the last equality follows from (27). In the same way, one proves that Θ^{4} = C_{12}...C_{n−1,n}, σ_{i}^{2} = C_{i,i+1}C_{i+1,i+2}C_{i,i+1}^{−1} and (Θ, σ_{i}) = (Ψ, σ_{i}) = 1 are preserved. ⊓ ⊔

5.2 Construction of morphisms \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n}^{k}) ⋊ S_{n} using an associator Φ_{λ}

Let us keep the notation of the previous section. Set

a_{2n}(λ) := −(2n + 1)B_{2n+2}λ^{2n+2}/(2n + 2)!, ỹ_{λ} := −\frac{adx}{e^{λadx} − 1}(y),
Ã_{λ} := Φ_{λ}(ỹ_{λ}, t)e^{λỹ_{λ}}Φ_{λ}(ỹ_{λ}, t)^{−1}
= e^{−λt/2}Φ_{λ}(−ỹ_{λ} − t, t)e^{λ(ỹ_{λ}+t)}Φ_{λ}(−ỹ_{λ} − t, t)^{−1}e^{−λt/2},
B̃_{λ} := e^{λt/2}Φ_{λ}(−ỹ_{λ} − t, t)e^{λx}Φ_{λ}(ỹ_{λ}, t)^{−1}

(the identity in the definition of A_{λ} follows from the hexagon relation).

Proposition 33. We have

Ã_{λ}^{12,3} = e^{λt̄_{12}/2}{Φ_{λ}}^{3,1,2}Ã_{λ}^{2,13}{Φ_{λ}}^{2,1,3}e^{λt̄_{12}/2} · {Φ_{λ}}^{3,2,1}Ã_{λ}^{1,23}{Φ_{λ}}^{1,2,3},
B̃_{λ}^{12,3} = e^{−λt̄_{12}/2}{Φ_{λ}}^{3,1,2}B̃_{λ}^{2,13}{Φ_{λ}}^{2,1,3}e^{−λt̄_{12}/2} · {Φ_{λ}}^{3,2,1}B̃_{λ}^{1,23}{Φ_{λ}}^{1,2,3},

(B̃_{λ}^{12,3}, e^{λt̄_{12}/2}{Φ_{λ}}^{3,1,2}Ã_{λ}^{2,13}{Φ_{λ}}^{2,1,3}e^{λt̄_{12}/2})
= (e^{−λt̄_{12}/2}{Φ_{λ}}^{3,1,2}B̃_{λ}^{2,13}{Φ_{λ}}^{2,1,3}e^{−λt̄_{12}/2}, Ã_{λ}^{12,3})
= {Φ_{λ}}^{3,2,1}e^{λt̄_{23}}{Φ_{λ}}^{1,2,3},

so the formulas of Proposition 31 (restricted to the generators A_{i}, B_{i}, σ_{i}, C_{jk}) induce a morphism \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n}^{k}) ⋊ S_{n} (here ˆ 𝔱̄_{1,n}^{k} is the degree completion of 𝔱̄_{1,n}^{k}).

Proof. In this proof, we shift the indices of the generators of 𝔱_{n+1} by 1, so these generators are now t_{ij}, i ≠ j ∈ {0, ..., n} (recall that 𝔱_{n+1} = 𝔱_{n+1}^{k}, t̄_{1,n} = 𝔱̄_{1,n}^{k}).

We have a morphism α_{n} : 𝔱_{n+1} → 𝔱̄_{1,n}, defined by t_{ij} ↦ t̄_{ij} if 1 ≤ i < j ≤ n and t_{0i} ↦ ỹ_{i} := −\frac{adx̄_{i}}{e^{λadx̄_{i}} −1}(ȳ_{i}) if 1 ≤ i ≤ n (it takes the central element ∑_{0≤i<j≤n} t_{ij} to 0).

Let ϕ : {1, ..., m} → {1, ..., n} be a map and ϕ′ : {0, ..., m} → {0, ..., n} be given by ϕ′(1) = 1, ϕ′(i) = ϕ(i) for i = 1, ..., m. The diagram

𝔱_{n+1} →^{x↦x^{ϕ′}} 𝔱_{m+1}
α_{n} ↓  ↓α_{m}
𝔱̄_{1,n} →^{x↦x^{ϕ}} 𝔱̄_{1,m}

is not commutative, we have instead the identity

α_{m}(x^{ϕ′}) = α_{n}(x)^{ϕ} − ∑_{i=1}^{n} ξ_{i}(x)( ∑_{i′,j′∈ϕ^{−1}(i)|i′<j′} t̄_{i′j′}),

where ξ_{i} : 𝔱̄_{1,n} → k is the linear form defined by ξ_{i}(t_{0i}) = 1, ξ_{i}(any other homogeneous Lie polynomial in the t_{kl}) = 0.

Since the various ∑_{i′,j′∈ϕ^{−1}(i)|i′<j′} t̄_{i′j′} commute with each other and with the image of x ↦ x^{ϕ}, this implies

α_{m}(g^{ϕ′}) = α_{n}(g)^{ϕ} ∏_{i=1}^{n} e^{−ξ_{i}(logg)(∑_{i′,j′∈ϕ^{−1}(i),i′<j′} t̄_{i′j′})}

for g ∈ exp(𝔱̂_{n+1}).

Set Ā_{λ} := Φ_{λ}^{0,1,2}e^{λt_{01}}(Φ_{λ}^{0,1,2})^{−1} ∈ exp(𝔱̂_{3}). One proves that

Ā_{λ}^{0,12,3}e^{λt_{12}} = e^{λt_{12}/2}Φ_{λ}^{3,1,2}Ā_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{λt_{12}/2} · Φ_{λ}^{3,2,1}Ā_{λ}^{0,1,23}Φ_{λ}^{1,2,3} (relation in exp(𝔱̂_{4})). We then have α_{2}(Ā_{λ}) = Ã_{λ}, α_{3}(Φ_{λ}^{1,2,3}) = Φ_{λ}^{1,2,3}, and the relation between the α_{i} and coproducts implies α_{3}(Ā_{λ}^{0,1,23}) = Ã_{λ}^{1,23} and α_{3}(Ā_{λ}^{0,12,3}e^{λt_{12}}) = Ã_{λ}^{12,3}. Taking the image by α_{3}, we get the first identity. As we have already mentioned, this identity implies (Φ_{λ}^{−1}Ã_{λ}^{1,23}Φ_{λ}, Ã_{λ}^{12,3}) = 1.

Let exp(𝔱̂_{n+1}) ∗ ℤ^{n}/I_{n} be the quotient of the free product of exp(𝔱̂_{n+1}) with ℤ^{n} = ⊕_{i=1}^{n}ℤX_{i} by the normal subgroup generated by the rations of the exponentials of the sides of each of the equations

X_{i}t_{0i}X_{i}^{−1} = ∑_{0≤α≤n,α≠i} t_{αi}, X_{i}(t_{0j} + t_{ij})X_{i}^{−1} = t_{0j},

X_{i}t_{jk}X_{i}^{−1} = t_{jk}, X_{j}X_{k}t_{jk}(X_{j}X_{k})^{−1} = t_{jk}

where i, j, k are distinct in {1, ..., n}. Then the morphism α_{n} : 𝔱_{n+1} → 𝔱̄_{1,n} extends to α̃_{n} : exp(𝔱̂_{n+1}) ∗ ℤ^{n}/I_{n} → exp(ˆ𝔱̄_{1,n}) by X_{i} ↦ e^{λx_{i}}.

If ϕ : {1, ..., m} → {1, ..., n} is a map, then the Lie algebra morphism 𝔱_{n+1} → 𝔱_{m+1}, x ↦ x^{ϕ′} extends to a group morphism exp(𝔱̂_{n}) ∗ ℤ^{n}/I_{n} → exp(𝔱̂_{m}) ∗ ℤ^{m}/I_{m} by X_{i} ↦ ∏_{i′∈ϕ^{−1}(i)} X_{i′}.

Let

B̄_{λ} := e^{λt_{12}/2}Φ_{λ}^{0,2,1}X_{1}Φ_{λ}^{2,1,0} ∈ exp(𝔱̂_{3}) ∗ ℤ^{2}/I_{2},

then α_{2}(B̄_{λ}) = B̃_{λ}.

We will prove that

B̄_{λ}^{0,12,3} = e^{−λt_{12}/2}Φ_{λ}^{3,1,2}B̄_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{−λt_{12}/2} · Φ^{3,2,1}B̄_{λ}^{0,1,23}Φ_{λ}^{1,2,3}.  (40)

The l.h.s. is

B̄_{λ}^{0,12,3} = e^{λt_{3,12}/2}Φ_{λ}^{0,3,12}X_{1}X_{2}Φ_{λ}^{3,21,0}

and the r.h.s. is

e^{−λt_{12}/2}Φ_{λ}^{3,1,2}e^{λt_{31,2}/2}Φ_{λ}^{0,13,2}X_{2}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}Φ_{λ}^{3,2,1}e^{λt_{23,1}/2}Φ_{λ}^{0,23,1}X_{1}Φ_{λ}^{32,1,0}Φ_{λ}^{1,2,3}.

The equality between these terms is rewritten as

X_{1}X_{2} = Φ_{λ}^{03,1,2}Φ_{λ}^{1,3,0}e^{−λt_{13}/2}X_{2}Φ_{λ}^{13,2,0}e^{λt_{13}/2}Φ_{λ}^{2,3,1}Φ_{λ}^{0,23,1}X_{1}Φ_{λ}^{01,2,3}Φ_{λ}^{2,1,0},

or, using the fact that X_{i} commutes with t_{jk} (i, j, k distinct), as

X_{1}X_{2} = Φ_{λ}^{03,1,2}Φ_{λ}^{1,3,0}X_{2}Φ_{λ}^{02,3,1}Φ_{λ}^{3,2,0}X_{1}Φ_{λ}^{01,2,3}Φ_{λ}^{2,1,0}.

Now X_{2}Φ_{λ}^{02,3,1} = Φ_{λ}^{0,3,1}X_{2}, X_{1}Φ_{λ}^{01,2,3} = Φ_{λ}^{0,2,3}X_{1} and X_{1}X_{2}Φ_{λ}^{2,1,0} = Φ_{λ}^{2,1,03}X_{1}X_{2}, so the r.h.s. is rewritten as Φ_{λ}^{03,1,2}Φ_{λ}^{1,3,0}Φ_{λ}^{0,3,1}X_{2}Φ_{λ}^{3,2,0}Φ_{λ}^{0,2,3}X_{1}Φ_{λ}^{2,1,0} = X_{1}X_{2}. This ends the proof of (40). Taking the image by α_{4}, we then get the second identity of the Proposition.

Let us prove the next identity. We have

218  D. Calaque, B. Enriquez and P. Etingof

(B̄_{λ}^{0,12,3}, e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Ā_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{λt̄_{12}/2})

= e^{λt_{12,3}/2}Φ_{λ}^{0,3,12}X_{1}X_{2}Φ_{λ}^{3,12,0}e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}e^{λt_{0,2}}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}e^{λt̄_{12}/2}Φ_{λ}^{0,12,3}

(X_{1}X_{2})^{−1}Φ_{λ}^{12,3,0}e^{−λt_{12,3}/2}e^{−λt̄_{12}/2}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}e^{−λt_{0,2}}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}e^{−λt̄_{12}/2}. Now

X_{1}X_{2}Φ_{λ}^{3,12,0}e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}e^{λt_{0,2}}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}e^{λt̄_{12}/2}Φ_{λ}^{0,12,3}(X_{1}X_{2})^{−1}
= e^{λt̄_{12}/2}X_{1}X_{2}Φ_{λ}^{3,12,0}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}e^{λt_{0,2}}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}Φ_{λ}^{0,12,3}(X_{1}X_{2})^{−1}e^{λt̄_{12}/2}
= e^{λt̄_{12}/2}X_{1}X_{2}Φ_{λ}^{0,2,1}Φ_{λ}^{3,1,02}e^{λt_{0,2}}Φ_{λ}^{02,1,3}Φ_{λ}^{0,2,1}(X_{1}X_{2})^{−1}e^{λt̄_{12}/2}
= e^{λt̄_{12}/2}X_{1}X_{2}Φ_{λ}^{0,2,1}e^{λt_{0,2}}Φ_{λ}^{0,2,1}(X_{1}X_{2})^{−1}e^{λt̄_{12}/2}
= e^{λt̄_{12}/2}Φ_{λ}^{03,2,1}X_{1}X_{2}e^{λt_{0,2}}(X_{1}X_{2})^{−1}Φ_{λ}^{03,2,1}e^{λt̄_{12}/2}
= e^{λt̄_{12}/2}Φ_{λ}^{03,2,1}e^{λt_{03,2}}Φ_{λ}^{03,2,1}e^{λt̄_{12}/2}.

Plugging this in the above expression for (B̄_{λ}^{0,12,3}, e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Ā_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{λt_{12}/2}), one then finds (B̄_{λ}^{0,12,3}, e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Ā_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{λt̄_{12}/2}) == Φ_{λ}^{3,2,1}e^{λt_{23}}Φ_{λ}^{1,2,3}. Taking the image by α_{4}, we then obtain (B̃_{λ}^{12,3}, e^{λt̄_{12}/2}Φ_{λ}^{3,1,2}Ã_{λ}^{2,13}Φ_{λ}^{2,1,3}e^{λt̄_{12}/2}) = Φ_{λ}^{3,2,1}e^{λt̄_{23}}Φ_{λ}^{1,2,3}.

Let us prove that last identity. For this, we will show

(e^{−λt_{12}/2}Φ_{λ}^{3,1,2}B̄_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}, Ā_{λ}^{0,12,3}e^{λt_{12}}) = Φ_{λ}^{3,2,1}e^{λt_{23}}Φ_{λ}^{1,2,3} and take the image by α_{4}.

We have

(e^{−λt_{12}/2}Φ_{λ}^{3,1,2}B̄_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}, Ā_{λ}^{0,12,3}e^{λt_{12}})

= e^{−λt_{12}/2}Φ_{λ}^{3,1,2}e^{λt_{2,13}/2}Φ_{λ}^{0,13,2}X_{2}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}Φ_{λ}^{0,12,3}e^{λt_{0,12}}Φ_{λ}^{3,12,0}e^{λt_{12}}e^{λt_{12}/2}

Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}X_{2}^{−1}Φ_{λ}^{2,13,0}e^{−λt_{2,13}/2}Φ_{λ}^{2,1,3}e^{λt_{12}/2}Φ_{λ}^{0,12,3}e^{−λt_{0,12}}Φ_{λ}^{3,12,0}e^{−λt_{12}}

= e^{−λt_{12}/2}Φ_{λ}^{3,1,2}e^{λt_{2,13}/2}Φ_{λ}^{0,13,2}X_{2}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}Φ_{λ}^{0,12,3}e^{λt_{0,12}+λt_{12}}Φ_{λ}^{3,12,0}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}

Φ_{λ}^{2,13,0}e^{−λt_{2,13}/2}X_{2}^{−1}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}Φ_{λ}^{0,12,3}e^{−λt_{0,12}}Φ_{λ}^{3,12,0}. Now

X_{2}Φ_{λ}^{13,2,0}Φ_{λ}^{2,1,3}Φ_{λ}^{0,12,3}e^{λt_{0,12}+λt_{12}}Φ_{λ}^{3,12,0}Φ_{λ}^{3,1,2}Φ_{λ}^{0,2,13}X_{2}^{−1}
= X_{2}Φ_{λ}^{02,1,3}Φ_{λ}^{1,2,0}e^{λt_{0,12}+λt_{12}}Φ_{λ}^{0,2,1}Φ_{λ}^{3,1,02}X_{2}^{−1}
= Φ_{λ}^{0,1,3}X_{2}Φ_{λ}^{1,2,0}e^{λt_{0,12}+λt_{12}}Φ_{λ}^{0,2,1}X_{2}^{−1}Φ_{λ}^{3,1,0}
= Φ_{λ}^{0,1,3}X_{2}e^{λ(t_{01}+t_{02}+t_{12})}X_{2}^{−1}Φ_{λ}^{3,1,0}
= Φ_{λ}^{0,1,3}e^{λ(t_{01}+t_{02}+t_{12}+t_{23})}Φ_{λ}^{3,1,0}.

So

(e^{−λt_{12}/2}Φ_{λ}^{3,1,2}B̄_{λ}^{0,2,13}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}, Ā_{λ}^{0,12,3}e^{λt_{12}})
= e^{−λt_{12}/2}Φ_{λ}^{3,1,2}e^{λt_{2,13}/2}Φ_{λ}^{0,13,2}Φ_{λ}^{0,1,3}e^{λ(t_{01}+t_{02}+t_{12}+t_{23})}
Φ_{λ}^{3,1,0}Φ_{λ}^{2,13,0}e^{−λt_{2,13}/2}Φ_{λ}^{2,1,3}e^{−λt_{12}/2}Φ_{λ}^{0,12,3}e^{−λt_{0,12}}Φ_{λ}^{3,12,0};

after some computation, we find that this equals Φ_{λ}^{3,2,1}e^{λt_{23}}Φ_{λ}^{1,2,3}. ⊓ ⊔

In particular, (Φ_{λ}, Ã_{λ}, B̃_{λ}) give rise to a morphism \overline{B}_{1,n} → exp(ˆ𝔱̄_{1,n}^{k}) ⋊ S_{n}; one proves as in Section 2 that it induces an isomorphism of filtered Lie algebras Lie(\overline{PB}_{1,n})_{k} ≃ ˆ 𝔱̄_{1,n}^{k}. Taking Φ_{λ} to be a rational associator [Dri91], we then obtain:

Corollary 34. We have a filtered isomorphism Lie(\overline{PB}_{1,n})_{ℚ} ≃ ˆ 𝔱̄_{1,n}^{ℚ}, which can

be extended to an isomorphism \overline{B}_{1,n}(ℚ) ≃ exp(ˆ𝔱̄_{1,n}^{ℚ}) ⋊ S_{n}.

5.3 Construction of morphisms Γ_{1,[n]} → G_{1,n} ⋊ S_{n} using a pair (Φ_{λ}, Θ̃_{λ})

Keep the notation of the previous section and set

Ψ̃_{λ} := exp(−\frac{1}{λ}(Δ_{0} + ∑_{k≥1} a_{2k}(λ)δ_{2k})).

Proposition 35. We have

[Ψ̃_{λ}]e^{λt̄_{12}/12}Ã_{λ}([Ψ̃_{λ}]e^{λt̄_{12}/12})^{−1} = Ã_{λ}, [Ψ̃_{λ}]e^{λt̄_{12}/12}B̃_{λ}([Ψ̃_{λ}]e^{λt̄_{12}/12})^{−1} = B̃_{λ}Ã_{λ}.

Proof. The first identity follows from the fact that Δ_{0} + ∑_{k≥1} a_{2k}(λ)[δ_{2k}] − λ^{2}t/12 commutes with t and ỹ_{λ}; the second identity follows from these facts and the analogue of Lemma 30, where 2πi is replaced by λ. ⊓ ⊔

Assume that Θ̃_{λ} ∈ G_{1} satisfies

Θ̃_{λ}^{4} = (Θ̃_{λ}Ψ̃_{λ})^{3} = (Θ̃_{λ}^{2}, Ψ̃_{λ}) = 1,

[Θ̃_{λ}]e^{λt̄_{12}/4}Ã_{λ}([Θ̃_{λ}]e^{λt̄_{12}/4})^{−1} = B̃_{λ}^{−1}, [Θ̃_{λ}]e^{λt̄_{12}/4}B̃_{λ}([Θ̃_{λ}]e^{λt̄_{12}/4})^{−1} = B̃_{λ}Ã_{λ}B̃_{λ}^{−1}

(one can show that the two last equations are equivalent), then Θ ↦ [Θ̃_{λ}]e^{λ(∑_{i<j} t̄_{ij})/4}, Ψ ↦ [Ψ̃_{λ}]e^{λ(∑_{i<j} t̄_{ij})/12} extends the morphism defined in Proposition 33 to a morphism Γ_{1,[n]} → G_{n} ⋊ S_{n}.

We do not know whether for each Φ_{λ} defined over k, there exists a Θ̃_{λ} defined over k, satisfying the above conditions.

5.4 Elliptic structures over QTQBA’s

Let (H, Δ_{H}, R_{H}, Φ_{H}) be a quasitriangular quasibialgebra (QTQBA). Recall that this means that [Dri90b]: (H, m_{H}) is an algebra, Δ_{H} : H → H^{⊗2} is an algebra morphism, R_{H} ∈ H^{⊗2} and Φ_{H} ∈ H^{⊗3} are invertible, and

Δ_{H}(x)^{2,1} = R_{H}Δ_{H}(x)R_{H}^{−1}, (id ⊗ Δ_{H}) ∘ Δ_{H}(x) = Φ_{H}(Δ_{H} ⊗ id) ∘ Δ_{H}(x)Φ_{H}^{−1}, R_{H}^{12,3} = Φ_{H}^{3,1,2}R_{H}^{1,3}(Φ_{H}^{1,3,2})^{−1}R_{H}^{2,3}Φ_{H}^{1,2,3}, R_{H}^{1,23} = (Φ_{H}^{2,3,1})^{−1}R_{H}^{1,3}Φ_{H}^{2,1,3}R_{H}^{1,2}(Φ_{H}^{1,2,3})^{−1},

Φ_{H}^{1,2,34}Φ_{H}^{12,3,4} = Φ_{H}^{2,3,4}Φ_{H}^{1,23,4}Φ_{H}^{1,2,3}.

One also assumes the existence of a unit 1_{H} and a counit ε_{H}.

If A is an algebra and J_{1}, J_{2} ⊂ A are left ideals, define the Hecke bimodule H(A|J_{1}, J_{2}) or H(J_{1}, J_{2}) as Hom_{A}(A/J_{1}, A/J_{2}) = (A/J_{2})^{J_{1}} where J_{1} acts on the quotient from the left; we have thus H(J_{1}, J_{2}) = {x ∈ A|J_{1}x ⊂ J_{2}}/J_{2}. The product of A induces a product H(J_{1}, J_{2})⊗H(J_{2}, J_{3}) → H(J_{1}, J_{3}). When J_{1} = J_{2} = J, H(J) := H(J, J) is the usual Hecke algebra, and H(J_{1}, J_{2}) is a (H(J_{1}), H(J_{2}))-bimodule. Recall that we have a functor A−mod → H(J)−mod, V ↦ V^{J} := {v ∈ V |Jv = 0}.

If H is an algebra with unit equipped with a morphism Δ_{H} : H → H^{⊗2} and a : H → D is a morphism of algebras with unit, we define for each n ≥ 1 and each pair of words w, w′ in the free magma generated by 1, ..., n containing 1, ..., n exactly once (recall that a magma is a set with a nonnecessarily associative binary operation) the Hecke bimodule

H^{w,w′}(D, H) := H(D ⊗ H^{⊗n}|J_{w}, J_{w′}),

(or simply H^{w,w′}) where J_{w} ⊂ D ⊗ H^{⊗n} is the left ideal generated by the image of (a ⊗ Δ_{H}^{w}) ∘ Δ_{H} : H_{+} → D ⊗ H^{⊗n}. Here H_{+} = Ker(H →^{ε_{H}} k) and for example Δ_{H}^{(21)3} = (213) ∘ (Δ_{H} ⊗ id_{H}) ∘ Δ_{H}, etc. We have products H^{w,w′} ⊗ H^{w′,w′′} → H^{w,w′′}. We denote the Hecke algebra H^{w,w} by H^{w}(D, H) or H^{w}; we denote by 1_{w} its unit. We denote by (H^{w,w′})^{×} the set of invertible elements of H^{w,w′}, i.e., the set of elements X such that for some X′ ∈ H^{w′,w}, X′X = 1_{w′}, XX′ = 1_{w}. The symmetric group S_{n} acts on the system of bimodules H^{w,w′} by permuting the factors, so we get maps Ad(σ) : H^{w,w′} → H^{σ(w),σ(w′)} (where σ(w) is the word w, where i is replaced by σ(i)). If w_{0} = ((12)...)n, we define an algebra structure on ⊕_{σ∈S_{n}}H^{w_{0},σ(w_{0})}σ by (∑_{σ∈S_{n}} h_{σ}σ)(∑_{τ∈S_{n}} h′_{τ} τ) := ∑_{σ,τ∈S_{n}} h_{σ}Ad(σ)(h′_{τ})στ. Then ⊔_{σ∈S_{n}}(H^{w_{0},σ(w_{0})})^{×}σ ⊂ ⊕_{σ∈S_{n}}H^{w_{0},σ(w_{0})}σ is a group with unit 1_{w_{0}}. We have an exact sequence 1 → (H^{w_{0}})^{×} → ⊔_{σ∈S_{n}}(H^{w_{0},σ(w_{0})})^{×}σ → S_{n}, but the last map is not necessarily surjective (and if it is, does not necessarily split).

If H is a quasibialgebra, then Φ_{H} gives rise to an element of H^{1(23),(12)3}(D, H), which we also denote Φ_{H}; similarly Φ_{H}^{−1} gives rise to the inverse (w.r.t. composition of Hecke bimodules) element Φ_{H}^{−1} ∈ H^{(12)3,1(23)}(D, H). We have algebra morphisms H^{12}(D, H) → H^{(12)3}(D, H) induced by X ↦ X^{0,12,3} := (id_{D} ⊗ Δ_{H} ⊗ id_{H})(X) (0 is the index of D) and similarly morphisms H^{12}(D, H) → H^{2(13)}(D, H), X ↦ X^{0,2,13}, H^{12}(D, H) → H^{1}(D, H), X ↦ X^{0,1,∅} and X^{0,∅,1}, etc. If moreover H is quasitriangular, then R_{H} ∈ H^{21,12}(D, H), R_{H}^{−1} ∈ H^{12,21}(D, H), so in that case ⊔_{σ∈S_{n}}H^{w_{0},σ(w_{0})}σ → S_{n} is surjective, and we have a morphism B_{n} → ⊔_{σ∈S_{n}}H^{w_{0},σ(w_{0})}σ such that the composition B_{n} → ⊔_{σ∈S_{n}}H^{w_{0},σ(w_{0})}σ → S_{n} is the canonical projection.

Definition 36. If H is a QTQBA, an elliptic structure on H is a triple (D, A, B), where D is an algebra with unit, equipped with an algebra morphism a : H → D, and A, B ∈ H^{12}(D, H) are invertible such that A^{0,1,∅} = A^{0,∅,1} = B^{0,1,∅} = B^{0,∅,1} = 1_{D} ⊗ 1_{H},

A^{0,12,3} = R_{H}^{2,1}(Φ_{H}^{2,1,3})^{−1}A^{0,2,13}Φ_{H}^{2,1,3}R_{H}^{1,2}(Φ_{H}^{1,2,3})^{−1}A^{0,1,23}Φ_{H}^{1,2,3},  (41)
B^{0,12,3} = (R_{H}^{1,2})^{−1}(Φ_{H}^{2,1,3})^{−1}B^{0,2,13}Φ_{H}^{2,1,3}(R_{H}^{2,1})^{−1}(Φ_{H}^{1,2,3})^{−1}B^{0,1,23}Φ_{H}^{1,2,3} (42)

and

(B^{0,12,3}, R_{H}^{2,1}(Φ_{H}^{2,1,3})^{−1}A^{0,2,13}Φ_{H}^{2,1,3}R_{H}^{1,2})

= ((R_{H}^{1,2})^{−1}(Φ_{H}^{2,1,3})^{−1}B^{0,2,13}Φ_{H}^{2,1,3}(R_{H}^{2,1})^{−1}, A^{0,12,3}) = (Φ_{H}^{1,2,3})^{−1}R_{H}^{3,2}R_{H}^{2,3}Φ_{H}^{1,2,3}

(identities in H^{(12)3}(D, H)).

The pair of identities (41), (42) is equivalent to

\begin{cases}R_{H}^{2,1}A^{0,2,1}R_{H}^{1,2}A^{0,1,2} = 1 \\ R_{H}^{3,12}A^{0,3,12}Φ_{H}^{3,1,2}R_{H}^{2,31}A^{0,2,31}Φ_{H}^{2,3,1}R_{H}^{1,23}A^{0,1,23}Φ_{H}^{1,2,3} = 1, \end{cases}

and

\begin{cases}(R_{H}^{1,2})^{−1}B^{0,2,1}(R_{H}^{2,1})^{−1}B^{0,1,2} = 1 \\ (R_{H}^{−1})^{12,3}B^{0,3,12}Φ_{H}^{3,1,2}(R_{H}^{−1})^{31,2}B^{0,2,31}Φ_{H}^{2,3,1}(R_{H}^{−1})^{23,1}B^{0,1,23}Φ_{H}^{1,2,3} = 1, \end{cases}

so the invertibility conditions on A, B follow from (41), (42).

If F ∈ H^{⊗2} is invertible with (ε_{H} ⊗ id_{H})(F) = (id_{H} ⊗ ε_{H})(F) = 1_{H}, then the twist of H by F is the quasi-Hopf algebra ^{F}H with product m_{H}, coproduct Δ̃_{H}(x) = F Δ_{H}(x)F^{−1}, R-matrix R̃_{H} = F^{2,1}R_{H}F^{−1} and associator Φ̃_{H} = F^{2,3}F^{1,23}Φ_{H}(F^{1,2}F^{12,3})^{−1}. If a : H → D is an algebra morphism, it can be viewed as a morphism ^{F}H → D, and we have an algebra isomorphism H^{(12)3}(D, H) → H^{(12)3}(D,^{F}H), induced by X ↦ F^{1,2}F^{0,12}X(F^{1,2}F^{0,12})^{−1} (more generally, we have an isomorphism of the systems of bimodules H^{w,w′}(D, H) → H^{w,w′}(D,^{F}H) induced by X ↦ F_{w}XF_{w′}^{−1} for suitable F_{w}).

If (D, A, B) is an elliptic structure on H, then an elliptic structure ^{F}H is (D, Ã, B̃), where Ã = F^{1,2}F^{0,12}A(F^{1,2}F^{0,12})^{−1} and B̃ = F^{1,2}F^{0,12}B(F^{1,2}F^{0,12})^{−1}.

An elliptic structure (D, A, B) over H gives rise to a unique group morphism

\overline{B}_{1,n} → ⊔_{σ∈S_{n}}H^{w_{0},σ(w_{0})}(D, H)^{×}σ,

such that

σ_{i} ↦ (Φ_{H}^{(((12)3)...i−1),i,i+1})^{−1}R_{H}^{i,i+1}(i, i + 1)Φ_{H}^{(((12)3)...i−1),i,i+1},
A_{i} ↦ Φ_{H,i}^{−1}A^{0,(((12)3)...i−1),(i...(n−1,n))}Φ_{H,i},
B_{i} ↦ Φ_{H,i}^{−1}B^{0,(((12)3)...i−1),(i...(n−1,n))}Φ_{H,i},

where

Φ_{H,i} = Φ_{H}^{((12)...i−1),i,(i+1(...(n−1,n)))}...Φ_{H}^{((12)...n−2),n−1,n};

here we have for example x^{((12)3)} = (Δ_{H} ⊗ id_{H}) ∘ Δ_{H}(x) for x ∈ H.

If 𝔤 is a Lie algebra and t_{𝔤} ∈ S^{2}(𝔤)^{𝔤} is nondegenerate, then H = U(𝔤)[[ℏ]] is a QTQBA, with m_{H}, Δ_{H} are the undeformed product and coproduct, R_{H} = e^{ℏt_{𝔤}/2} and Φ_{H} = Φ(ℏt_{𝔤}^{1,2}, ℏt_{𝔤}^{2,3}), where Φ is an 1-associator. The results of next Section then imply that (D, A, B) is an elliptic structure over H, where D = D(𝔤)[[ℏ]] (D(𝔤) is the algebra of algebraic differential operators on 𝔤) and A, B are given by the formulas for Ã_{λ}, B̃_{λ} with t replaced by ℏt_{𝔤}^{1,2}, x replaced by ℏ ∑_{α} x_{α} ⊗ (e_{α}^{1} + e_{α}^{2}), y replaced by ℏ ∑_{α} ∂_{α} ⊗ (e_{α}^{1} + e_{α}^{2}).

Remark 37. If H is a Hopf algebra, we have an isomorphism

H^{w_{0}}(D, H) ≃ (D ⊗ H^{⊗n−1})^{H},

where the right side is the commutant of the diagonal map H → D ⊗ H^{⊗n−1}, h ↦ (a ⊗ id_{H}^{⊗n−1}) ∘ Δ_{H}^{(n)}(h). This map takes the class of d ⊗ h_{1} ⊗ ... ⊗ h_{n} to da(S_{H}(h_{n}^{(n)})) ⊗ h_{1}S_{H}(h_{n}^{(n−1)}) ⊗ ... ⊗ h_{n−1}S_{H}(h_{n}^{(1)}) (S_{H} is the antipode of H). So A, B identify with elements A, B ∈ (D ⊗ H)^{H}; the conditions are then

A^{0,12} = R_{H}^{2,1}A^{0,2}R_{H}^{1,2}A^{0,1}, B^{0,12} = (R_{H}^{1,2})^{−1}B^{0,2}(R_{H}^{2,1})^{−1}B^{0,1},

(B^{0,12}, R_{H}^{2,1}A^{0,2}R_{H}^{1,2}) = ((R_{H}^{1,2})^{−1}B^{0,2}(R_{H}^{2,1})^{−1}, A^{0,12})
= (R_{H}^{3,2}R_{H}^{1,2}R_{H}^{0,2}R_{H}^{2,0}R_{H}^{2,1}R_{H}^{2,3})^{˜0,˜1,2·˜3}

(conditions in (D ⊗H^{⊗2})^{H}), where the superscript B′_{n} ⋊ℤ^{n−1} → B_{n−1} ⋊ℤ^{n−1} is the map x_{0} ⊗ ... ⊗ x_{3} ↦ S_{H}(x_{0}) ⊗ S_{H}(x_{1}) ⊗ x_{2}S_{H}(x_{3}).

Moreover, the morphism PB_{n} → (H^{w_{0}})^{×} ≃ (D ⊗ H^{⊗n−1})^{H} factors through PB_{n} → PB_{n−1} × ℤ^{n−1} → (D ⊗ H^{⊗n−1})^{H}, where: (a) the first morphism is induced by ℤ^{n−1} ⋊ B′_{n} → ℤ^{n−1} ⋊ B_{n−1} (where B′_{n} = B_{n} ×_{S_{n}} S_{n−1} is the group of braids leaving the last strand fixed), constructed as follows: we have a composition B′_{n+1} → π_{1}((ℙ^{1})^{n+1} − diagonals/S_{n}) → π_{1}(ℂ^{n} − diagonals/S_{n}) = B_{n}, where the first map is induced by ℂ ⊂ ℙ^{1}, and the middle map comes from the fibration ℂ^{n} − diagonals → (ℙ^{1})^{n+1} − diagonals → ℙ^{1}, (z_{1}, ..., z_{n}) → (z_{1}, ..., z_{n}, ∞) and (z_{1}, ..., z_{n+1}) → z_{n+1} [the second projection has a section so the map between π_{1}’s is an isomorphism]; viewing ℤ^{n−1} ⋊B′_{n}, ℤ^{n−1} ⋊B_{n−1} as fundamental groups of configuration spaces of points equipped with a nonzero tangent vector, we then get the morphism ℤ^{n−1} ⋊ B′_{n} → ℤ^{n−1} ⋊ B_{n−1} (which does not restrict to a morphism B′_{n} → B_{n−1}); (b) the second map is induced by the standard map PB_{n−1} × ℤ^{n−1} → (H^{⊗n−1})^{×} induced by R_{H} = ∑_{α} r′_{α} ⊗ r′′_{α} and the map taking the ith generator of ℤ^{n−1} to 1 ⊗ ... ⊗ uS_{H}(u) ⊗ ... ⊗ 1, where u = ∑_{i} S_{H}(r′′_{α})r′_{α} (see [Dri90a]). The morphism B_{n} → Aut((H^{w_{0}})^{×}) = Aut((D ⊗ H^{⊗n−1})^{H}) extends the inner action of PB_{n} by

σ_{n−1} · X := \{R_{H}^{n−1,n...2n−1}X^{0,1,...,n−2,n...2n−1}R_{H}^{n...2n−1,n−1}\}^{0·{ \atop 2n−1},...,n−1·ñ} (where the superscript means that x_{0} ⊗ ... ⊗ x_{2n−1} maps to x_{0}S_{H}(x_{2n−1}) ⊗ ... ⊗ x_{n−1}S_{H}(x_{n})).

We have then ⊔_{σ∈S_{n}}(H^{w_{0},σ(w_{0})})^{×}σ ≃ ((D ⊗ H^{⊗n−1})^{×})^{H} ⋊_{PB_{n}} B_{n} (the index means that PB_{n} ⊂ B_{n} is identified with its image in ((D⊗H^{⊗n−1})^{×})^{H}).

Then if (A, B) is an elliptic structure over a : H → D, the morphism B_{n} → ((D ⊗ H^{⊗n−1})^{×})^{H} ⋊_{PB_{n}} B_{n} extends to a morphism

\overline{B}_{1,n} → ((D ⊗ H^{⊗n−1})^{×})^{H} ⋊_{PB_{n}} B_{n}

via A_{i} ↦ A^{0,1...i−1}, B_{i} ↦ B^{0,1...i−1}.

This interpretation of H^{w_{0}} and of the relations between A, B can be extended to the case when H is a quasi-Hopf algebra.

Remark 38. Let C be a rigid braided monoidal category. We define an elliptic structure on C as a quadruple (E, A, B, F), where E is a category, F : E → C is a functor, and A, B are functorial automorphisms of F (?)⊗?, which reduce to the identity if the second factor is the neutral object 1, and such that the following equalities of automorphisms of F (M)⊗(X ⊗Y) hold (we write them omitting associativity maps, as they can be put in automatically):

A_{M,X⊗Y} = β_{Y,X}A_{M,Y} β_{X,Y} A_{M,X},
B_{M,X⊗Y} = β_{X,Y}^{−1} B_{M,Y} β_{Y,X}^{−1}B_{M,X},

(B_{M,X⊗Y} , β_{Y,X}A_{M,Y} β_{X,Y}) = (β_{Y,X}^{−1}B_{M,Y} β_{X,Y}^{−1} , A_{M,X⊗Y})
= β_{(M⊗X⊗Y)^{∗},Y} β_{Y,(M⊗X⊗Y)^{∗}} ∘ can_{M⊗X⊗Y},

where can_{X} ∈ Hom_{C}(1, X ⊗ X^{∗}) is the canonical map and the r.h.s. of the last identity is viewed as an element of End_{C}(M⊗X⊗Y) using its identification with Hom_{C}(1, (M⊗X⊗Y)⊗(M⊗X⊗Y)^{∗}). An elliptic structure on a quasitriangular quasi-Hopf algebra H gives rise to an elliptic structure on H-mod. An elliptic structure over a rigid braided monoidal category C gives rise to representations of \overline{B}_{1,n} by C-automorphisms of F (M) ⊗ X^{⊗n−1}.

6 The KZB connection as a realization of the universal KZB connection

6.1 Realizations of 𝔱̄_{1,n}

Let 𝔤 be a Lie algebra and t_{𝔤} ∈ S^{2}(𝔤)^{𝔤} be nondegenerate. We denote by (a, b) ↦ ⟨a, b⟩ the corresponding invariant pairing.

Let D(𝔤) be the algebra of algebraic differential operators on 𝔤. It has generators x_{a}, ∂_{a}, a ∈ 𝔤, and relations: a ↦ x_{a}, a ↦ ∂_{a} are linear, [x_{a}, x_{b}] = [∂_{a}, ∂_{b}] = 0, [∂_{a}, x_{b}] = ⟨a, b⟩.

There is a unique Lie algebra morphism 𝔤 → D(𝔤), a ↦ X_{a}, where X_{a} := ∑_{α} x_{[a,e_{α}]}∂_{e_{α}}, and t_{𝔤} = ∑_{α} e_{α} ⊗ e_{α} (it is the infinitesimal of the adjoint action). We also have a Lie algebra morphism 𝔤 → A_{n} := D(𝔤) ⊗ U(𝔤)^{⊗n}, a ↦ Y_{a} := X_{a} ⊗ 1 + 1 ⊗ (∑_{i=1}^{n} a^{(i)}). We denote by 𝔤^{diag} the image of this morphism. We denote by H_{n}(𝔤) the Hecke algebra of (A_{n}, 𝔤^{diag}). It is defined as the quotient {x ∈ A_{n}|∀a ∈ 𝔤, Y_{a}x ∈ A_{n}𝔤^{diag}}/A_{n}𝔤^{diag}. We have a natural action of S_{n} on A_{n}, which induces an action of S_{n} on H_{n}(𝔤).

If (V_{i})_{i=1,...,n} are 𝔤-modules, then (S(𝔤) ⊗ (⊗_{i=1}^{n}V_{i}))^{𝔤} is a module over H_{n}(𝔤). If moreover V_{1} = ... = V_{n}, this is a module over H_{n}(𝔤) ⋊ S_{n}.

Proposition 39. There is a unique Lie algebra morphism ρ_{𝔤} : 𝔱̄_{1,n} → H_{n}(𝔤), x̄_{i} ↦ ∑_{α} x_{α} ⊗ e_{α}^{(i)}, ȳ_{i} ↦ − ∑_{α} ∂_{α} ⊗ e_{α}^{(i)}, t̄_{ij} ↦ 1 ⊗ t_{𝔤}^{(ij)} (we set x_{α} := x_{e_{α}}, ∂_{α} := ∂_{e_{α}}).

Proof. The images of all the generators of 𝔱̄_{1,n} are contained in the commutant of 𝔤^{diag} in A_{n}, therefore also in its normalizer. According to Lemma 4, we will use the following presentation of 𝔱̄_{1,n}. Generators are x̄_{i}, ȳ_{i}, t̄_{ij}, relations are [x̄_{i}, x̄_{j}] = [ȳ_{i}, ȳ_{j}] = 0, [x̄_{i}, ȳ_{j}] = t̄_{ij} (i ≠ j), t̄_{ij} = t̄_{ji}, ∑_{i} x̄_{i} = ∑_{i} ȳ_{i} = 0, [x̄_{i}, t̄_{jk}] = [ȳ_{i}, t̄_{jk}] = 0 (i, j, k distinct).

The relations [x̄_{i}, x̄_{j}] = [ȳ_{i}, ȳ_{j}] = 0, [x̄_{i}, ȳ_{j}] = t̄_{ij} (i ≠ j), t̄_{ij} = t̄_{ji} and [x̄_{i}, t̄_{jk}] = [ȳ_{i}, t̄_{jk}] = 0 are obviously preserved. Let us check that ∑_{i} x̄_{i} = ∑_{i} ȳ_{i} = 0 are preserved.

We have

∑_{i} ρ_{𝔤}(x̄_{i}) = ∑_{α} x_{α} ⊗ (∑_{i} e_{α}^{(i)}) = ∑_{α}(x_{α} ⊗ 1)(Y_{α} − X_{α} ⊗ 1)
≡ − ∑_{α} x_{α}X_{α} ⊗ 1 = ∑_{α,β} x_{e_{α}}x_{[e_{α},e_{β}]}∂_{e_{β}} ⊗ 1 = 0

since x_{α} commutes with x_{[e_{α},e_{β}]} and ∑_{β} e_{β} ⊗ e_{β} = t_{𝔤} is invariant. We also have

∑_{i} ρ_{𝔤}(ȳ_{i}) = − ∑_{α} ∂_{α} ⊗ (∑_{i} e_{α}^{(i)}) = − ∑_{α}(∂_{α} ⊗ 1)(Y_{α} − X_{α} ⊗ 1)
≡ ∑_{α} ∂_{α}X_{α} ⊗ 1 = − ∑_{α,β} ∂_{e_{α}}x_{[e_{α},e_{β}]}∂_{e_{β}}
= − ∑_{α,β}⟨e_{α}, [e_{α}, e_{β}]⟩∂_{e_{β}} − ∑_{α,β} x_{[e_{α},e_{β}]}∂_{e_{α}}∂_{e_{β}};

since t_{𝔤} is invariant and ⟨−, −⟩ is symmetric, we have ∑_{α}⟨e_{α}, [e_{α}, e_{β}]⟩ = 0 for any β, and since [∂_{e_{α}}, ∂_{e_{β}}] = 0, we have ∑_{α,β} x_{[e_{α},e_{β}]}∂_{e_{α}}∂_{e_{β}}, so ∑_{i} ρ_{𝔤}(ȳ_{i}) = 0.

⊓ ⊔

6.2 Realizations of 𝔱̄_{1,n} ⋊ 𝔡

Let (𝔤, t_{𝔤}) be as in Subsection 6.1. We keep the same notations.

Proposition 40. The Lie algebra morphism ρ_{𝔤} : 𝔱̄_{1,n} → H_{n}(𝔤) of Proposition 39 extends to a Lie algebra morphism 𝔱̄_{1,n} ⋊ 𝔡 → H_{n}(𝔤), defined by Δ_{0} ↦ −\frac{1}{2}(∑_{α} ∂_{α}^{2}) ⊗ 1, X ↦ \frac{1}{2}(∑_{α} x_{α}^{2}) ⊗ 1, d ↦ \frac{1}{2}(∑_{α} x_{α}∂_{α} + ∂_{α}x_{α}) ⊗ 1, and

δ_{2m} → \frac{1}{2} ∑_{α_{1},...,α_{2m},α} x_{α_{1}} ⋯ x_{α_{2m}} ⊗ (∑_{i=1}^{n}(ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α}) · e_{α})^{(i)})

for m ≥ 1. This morphism further extends to a morphism U(𝔱̄_{1,n} ⋊ 𝔡) ⋊ S_{n} → H_{n}(𝔤) ⋊ S_{n} by σ ↦ σ.

Proof. First of all [ρ_{𝔤}(δ_{2m}), ρ_{𝔤}(x̄_{i})] equals

\frac{1}{2} ∑_{α_{1},...,α_{2m},α,β} x_{α_{1}} ⋯ x_{α_{2m}}x_{β} ⊗ [e_{β}, ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α})e_{α}]^{(i)}

= \frac{1}{2} ∑_{α_{1},...,α_{2m},α,β} x_{α_{1}} ⋯ x_{α_{2m}}x_{β} ⊗ ∑_{ℓ=1}^{2m} (ad(e_{α_{1}}) ⋯ ad([e_{β}, e_{α_{ℓ}}]) ⋯ ad(e_{α_{2m}})(e_{α})e_{α})^{(i)}

(the equality follows from the invariance of t_{𝔤}) which equals zero since the first factor is symmetric in (β, α_{ℓ}) while the second is antisymmetric in (β, α_{l}).

ρ_{𝔤} preserves the relation [δ_{2m}, t̄_{ij}] = [t̄_{ij}, ad(x̄_{i})^{2m}(t̄_{ij})], because ρ_{𝔤}(δ_{2m} + ∑_{i<j} ad(x̄_{i})^{2m}(t̄_{ij})) belongs to D(𝔤) ⊗ Im(Δ^{(n)} : U(𝔤) → U(𝔤)^{⊗n}), where Δ^{(n)} is the n-fold coproduct and U(𝔤) is equipped with its standard bialgebra structure.

Now [ρ_{𝔤}(δ_{2m}), ρ_{𝔤}(ȳ_{i})] yields

\frac{1}{2} ∑_{α_{1},...,α_{2m},α,β} ( ∑_{j}[∂_{β}, x_{α_{1}} ⋯ x_{α_{2m}}] ⊗ e_{β}^{(i)}ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α})^{(j)}e_{α}^{(j)}

+x_{α_{1}} ⋯ x_{α_{2m}}∂_{β} ⊗ [e_{β}, ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α}) · e_{α}]^{(i)})

= \frac{1}{2} ∑_{l=1}^{2m} ∑_{α_{1},...,α_{2m},α} ( ∑_{j} x_{α_{1}} ⋯ x̌_{α_{l}} ⋯ x_{α_{2m}} ⊗ e_{α_{l}}^{(i)}ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α})^{(j)}e_{α}^{(j)}

+x_{α_{1}} ⋯ x_{α_{2m}}∂_{β} ⊗ ad(e_{α_{1}}) ⋯ ad([e_{β}, e_{α_{l}}]) ⋯ ad(e_{α_{2m}})(e_{α})^{(i)}e_{α}^{(i)})

≡ \frac{1}{2} ∑_{l=1}^{2m} ∑_{α_{1},...,α_{2m},α} ∑_{j} (x_{α_{1}} ⋯ x̌_{α_{l}} ⋯ x_{α_{2m}} ⊗ e_{α_{l}}^{(i)}ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α})^{(j)}e_{α}^{(j)}

−x_{α_{1}} ⋯ x̌_{α_{l}} ⋯ x_{α_{2m}} ⊗ ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α})^{(i)}e_{α}^{(i)}e_{α_{l}}^{(j)}).

The term corresponding to j = i is

\frac{1}{2} ∑_{l=1}^{2m} ∑_{α_{1},...,α_{2m},α} x_{α_{1}} ⋯ x̌_{α_{l}} ⋯ x_{α_{2m}} ⊗ [e_{α_{l}}, ad(e_{α_{1}}) ⋯ ad(e_{α_{2m}})(e_{α}) · e_{α}]^{(i)}

It corresponds to the linear map S^{2m−1}(𝔤) → U(𝔤), such that for x ∈ 𝔤,

x^{2m−1} ↦ \frac{1}{2} ∑_{p+q=2m−1} ∑_{α,β}[e_{β}, ad(x)^{p}ad(e_{β})ad(x)^{q}(e_{α}) · e_{α}]

= \frac{1}{2} ∑_{α,β} ∑_{p+q+r=2m−2} ad(x)^{p}ad([e_{β}, x])ad(x)^{q}ad(e_{β})ad(x)^{r}(e_{α}) · e_{α}
+ad(x)^{p}ad(e_{β})ad(x)^{q}ad([e_{β}, x])ad(x)^{r}(e_{α}) · e_{α}

since μ(t_{𝔤}) = 0 (μ : 𝔤^{⊗2} → 𝔤 is the Lie bracket) and t_{𝔤} is 𝔤-invariant. Now this is zero since t_{𝔤} = ∑_{β} e_{β} ⊗ e_{β} is invariant.

The term corresponding to j ≠ i corresponds to the map S^{2m−1}(𝔤) → U(𝔤)^{⊗n}, such that for x ∈ 𝔤

x^{2m−1} ↦ −\frac{1}{2} ∑_{l=1}^{2m} ∑_{α,β} ((adx)^{l−1}(ade_{β})(adx)^{2m−l}(e_{α}) · e_{α})^{(i)}e_{β}^{(j)} − (i ↔ j)

= \frac{1}{2} ∑_{l=1}^{2m}(−1)^{l+1} ∑_{α,β} ((adx)^{l−1}([e_{β}, e_{α}]) · (adx)^{2m−l}(e_{α}))^{(i)}e_{β}^{(j)} − (i ↔ j)

= \frac{1}{2} ∑_{l=1}^{2m}(−1)^{l−1} ∑_{α,β} ((adx)^{l−1}(e_{β}) · (adx)^{2m−l}(e_{α}))^{(i)}[e_{α}, e_{β}]^{(j)} − (i ↔ j)

= \frac{1}{2} ∑_{l=1}^{2m}(−1)^{l}[ ∑_{α} ((adx)^{l−1}(e_{α}))^{(i)}e_{α}^{(j)}, ∑_{β} ((adx)^{2m−l}(e_{β}))^{(i)}e_{β}^{(j)}],

which coincides with the image of \frac{1}{2} ∑_{p+q=2m−1}(−1)^{q}[(adx̄_{i})^{p}(t̄_{ij}), (adx̄_{i})^{q}(t̄_{ij})].

It is then clear that ρ_{𝔤} preserves the commutation relations of Δ_{0}, X and d with δ_{2m}. ⊓ ⊔

6.3 Reductions

Assume that 𝔤 is finite dimensional and we have a reductive decomposition 𝔤 = 𝔥 ⊕ 𝔫, i.e., 𝔥 ⊂ 𝔤 is a Lie subalgebra and 𝔫 ⊂ 𝔤 is a vector subspace such that [𝔥, 𝔫] ⊂ 𝔫; assume also that t_{𝔤} = t_{𝔥} + t_{𝔫}, where t_{𝔥} ∈ S^{2}(𝔥)^{𝔥} and t_{𝔫} ∈ S^{2}(𝔫)^{𝔥}.

We assume that for a generic h ∈ 𝔥, ad(h)_{|𝔫} ∈ End(𝔫) is invertible. This condition is equivalent to the nonvanishing of P (λ) := det(ad(λ^{∨})_{|𝔫}) ∈ S^{dim𝔫}(𝔥), where λ ↦ λ^{∨} is the map 𝔥^{∗} → 𝔥, with λ^{∨} := (λ ⊗ id)(t_{𝔥}). If G is a Lie group with Lie algebra 𝔤, an equivalent condition is that a generic element of 𝔤^{∗} is conjugate to some element in 𝔥^{∗} (see [EE05]).

Let us set, for λ ∈ 𝔥^{∗},

r(λ) := (id ⊗ (adλ^{∨})_{|𝔫}^{−1})(t_{𝔫}),

Then r : 𝔥_{reg}^{∗} → ∧^{2}(𝔫) is an 𝔥-equivariant map (here 𝔥_{reg}^{∗} = {λ ∈ 𝔥^{∗}|P (λ) ≠ 0}), satisfying the classical dynamical Yang-Baxter (CDYB) equation

CYB(r) − Alt(dr) = 0

(see [EE05]). Here for r = ∑_{α} a_{α} ⊗ b_{α} ⊗ ℓ_{α} ∈ (𝔫^{⊗2} ⊗ S(𝔥)[1/P])^{𝔥}, we set CYB(r) = ∑_{α,α′}([a_{α}, a_{α′}] ⊗ b_{α} ⊗ b_{α′} + a_{α} ⊗ [b_{α}, a_{α′}] ⊗ b_{α′} + a_{α} ⊗ a_{α′} ⊗ [b_{α}, b_{α′}]) ⊗ ℓ_{α}ℓ_{α′}, dr := ∑_{α} a_{α} ⊗ b_{α} ⊗ dℓ_{α}, where d extends S(𝔥) → 𝔥 ⊗ S(𝔥), x^{k} ↦ kx ⊗ x^{k−1} and Alt(X ⊗ ℓ) = (X + X^{2,3,1} + X^{3,1,2}) ⊗ ℓ.

We also set

ψ(λ) := (id ⊗ (adλ^{∨})_{|𝔫}^{−2})(t_{𝔫}).

We write ψ(λ) = ∑_{α} A_{α} ⊗ B_{α} ⊗ L_{α}.

Let D(𝔥)[1/P] be the localization at P of the algebra D(𝔥) of differential operators on 𝔥; the latter algebra is generated by x̄_{h}, ¯∂_{h}, h ∈ 𝔥, with relations h ↦ x̄_{h}, h ↦ ¯∂_{h} linear, [x̄_{h}, x̄_{h′}] = [¯∂_{h}, ¯∂_{h′}] = 0, and [¯∂_{h}, x̄_{h′}] = ⟨h, h′⟩.

Set B_{n} := D(𝔥)[1/P]⊗U(𝔤)^{⊗n}. For h ∈ 𝔥, we define X̄_{h} := ∑_{ν} x̄_{[h,h_{ν}]} ¯∂_{h_{ν}} ∈ D(𝔥), where t_{𝔥} = ∑_{ν} h_{ν} ⊗ h_{ν}. We then set Ȳ_{h} := X̄_{h} + ∑_{i=1}^{n} h^{(i)}. The map 𝔥 → B_{n} is a Lie algebra morphism; we denote by 𝔥^{diag} its image.

We denote by H_{n}(𝔤, 𝔥) the Hecke algebra of B_{n} relative to 𝔥^{diag}. Explicitly, H_{n}(𝔤, 𝔥) = {x ∈ B_{n}|∀h ∈ 𝔥, Ȳ_{h}x ∈ B_{n}𝔥^{diag}}/B_{n}𝔥^{diag}.

Proposition 41. There is a unique Lie algebra morphism

ρ_{𝔤,𝔥} : 𝔱̄_{1,n} → H_{n}(𝔤, 𝔥),

such that x̄_{i} ↦ ∑_{ν} x̄_{ν} ⊗ h_{ν}^{(i)}, ȳ_{i} ↦ − ∑_{ν} ¯∂_{ν} ⊗ h_{ν}^{(i)} + ∑_{j} ∑_{α} ℓ_{α} ⊗ a_{α}^{(i)}b_{α}^{(j)},

t̄_{ij} ↦ t_{𝔤}^{(ij)}. Here r(λ) = ∑_{α} ℓ_{α}(λ)(a_{α} ⊗ b_{α}).

If V_{1}, ..., V_{n} are 𝔤-modules, then S(𝔥)[1/P] ⊗ (⊗_{i}V_{i}) is a module over D(𝔥)[1/P] ⊗ U(𝔤)^{⊗n}, and (S(𝔥)[1/P] ⊗ (⊗_{i}V_{i}))^{𝔥} is a module over H_{n}(𝔤, 𝔥).

Moreover, we have a restriction morphism (S(𝔤)⊗(⊗_{i}V_{i}))^{𝔤} → (S(𝔥)[1/P]⊗ (⊗V_{i}))^{𝔥}. Note that (S(𝔤)⊗(⊗_{i}V_{i}))^{𝔤} is a 𝔱̄_{1,n}-module using the morphism 𝔱̄_{1,n} → H_{n}(𝔤), while (S(𝔥)[1/P]⊗(⊗V_{i}))^{𝔥} is a 𝔱̄_{1,n}-module using the morphism 𝔱̄_{1,n} → H_{n}(𝔤, 𝔥). Then one checks that the restriction morphism (S(𝔤) ⊗ (⊗_{i}V_{i}))^{𝔤} → (S(𝔥)[1/P] ⊗ (⊗V_{i}))^{𝔥} is a 𝔱̄_{1,n}-modules morphism.

Proof. The images of the above elements are all 𝔥-invariant. To lighten the notation, we will imply summation over repeated indices and denote elements of B_{n} as follows: ¯∂_{ν} ⊗1 by ¯∂_{ν}, x̄_{ν} ⊗1 by ⟨λ, h_{ν}⟩, 1⊗x^{(i)} by x^{i}. Then ρ_{𝔤,𝔥}(x̄_{i}) = (λ^{∨})^{i}, ρ_{𝔤,𝔥}(ȳ_{i}) = −h_{ν}^{i} ¯∂_{ν} +∑_{j=1}^{n} r(λ)^{ij} (here for x⊗y ∈ 𝔤^{⊗2}, (x⊗y)^{ii} := x^{i}y^{i}).

We will use the same presentation of 𝔱̄_{1,n} as in Proposition 39. The relations [x̄_{i}, x̄_{j}] = 0 and t̄_{ij} = t̄_{ji} are obviously preserved.

Let us check that [x̄_{i}, ȳ_{j}] = t̄_{ij} is preserved (i ≠ j):

[ρ_{𝔤,𝔥}(x̄_{i}), ρ_{𝔤,𝔥}(ȳ_{j})] = [x̄_{ν}h_{ν}^{i}, −h_{ν}^{j} ¯∂_{ν} + ∑_{k} r(λ)^{jk}] = t_{𝔥}^{ij} + [λ^{i}, r(λ)^{ji}]
= t_{𝔥}^{ij} + t_{𝔫}^{ij} = t_{𝔤}^{ij} = ρ_{𝔤,𝔥}(t̄_{ij}).

Let us check that ∑_{i} x̄_{i} = ∑_{i} ȳ_{i} = 0 are preserved. We have ∑_{i} ρ_{𝔤,𝔥}(x̄_{i}) = 0 by the same argument as above and ∑_{i} ρ_{𝔤,𝔥}(ȳ_{i}) = ∑_{i}(λ^{∨})^{i} (by the antisymmetry of r(λ)), which vanishes by the same argument as above.

Let us check that [ȳ_{i}, ȳ_{j}] = 0 is preserved, for i ≠ j. We have

[ρ_{𝔤,𝔥}(ȳ_{i}), ρ_{𝔤,𝔥}(ȳ_{j})] = ∑_{k|k≠i,j} ( − h_{ν}^{i}(∂_{ν}r(λ))^{jk} + h_{ν}^{j}(∂_{ν}r(λ))^{ik} + [r(λ)^{ij}, r(λ)^{jk}]
+[r(λ)^{ik}, r(λ)^{jk}] + [r(λ)^{ik}, r(λ)^{ji}]) + [(h_{ν}^{i} + h_{ν}^{j})¯∂_{ν}, r(λ)^{ij}]
−[h_{ν}^{i} ¯∂_{ν}, r(λ)^{jj}] + [h_{ν}^{j} ¯∂_{ν}, r(λ)^{ii}] + [r(λ)^{ij}, r(λ)^{ii} + r(λ)^{jj}]
= ∑_{k|k≠i,j} h_{ν}^{k}(∂_{ν}r(λ))^{ij} + [(h_{ν}^{i} + h_{ν}^{j})¯∂_{ν}, r(λ)^{ij}] − [h_{ν}^{i} ¯∂_{ν}, r(λ)^{jj}]
+[h_{ν}^{j} ¯∂_{ν}, r(λ)^{ii}] + [r(λ)^{ij}, r(λ)^{ii} + r(λ)^{jj}]
≡ (∂_{ν}r(λ))^{ij}(−h_{ν}^{i} − h_{ν}^{j} − X̄_{ν}) + [(h_{ν}^{i} + h_{ν}^{j})¯∂_{ν}, r(λ)^{ij}]
−h_{ν}^{i}(∂_{ν}r(λ))^{jj} + h_{ν}^{j}(∂_{ν}r(λ))^{ii} + [r(λ)^{ij}, r(λ)^{ii} + r(λ)^{jj}]
= [h_{ν}^{i} + h_{ν}^{j}, r(λ)^{ij}]¯∂_{ν} − (∂_{ν}r^{ij}(λ))X̄_{ν} + [h_{ν}^{i} + h_{ν}^{j}, ∂_{ν}r(λ)^{ij}]
−h_{ν}^{i}(∂_{ν}r(λ))^{jj} + h_{ν}^{j}(∂_{ν}r(λ))^{ii} + [r(λ)^{ij}, r(λ)^{ii} + r(λ)^{jj}].

The second equality follows from the CDYBE and the antisymmetry on r(λ). Then

[h_{ν}^{i}+h_{ν}^{j}, r(λ)^{ij}]¯∂_{ν}−(∂_{ν}r^{ij}(λ))X̄_{ν} = ([h_{ν′}^{i}+h_{ν′}^{j}, r(λ)^{ij}]−∂_{ν}r^{ij}(λ)⟨λ, [h_{ν}, h_{ν′}]⟩)¯∂_{ν′}

is zero thanks to the 𝔥-invariance of r(λ). Applying x^{i}y^{j}z^{k} ↦ x^{i}(yz)^{i} to the CDYB identity

[r(λ)^{ij}, r(λ)^{ik}]+[r(λ)^{ij}, r(λ)^{jk}]+[r(λ)^{ik}, r(λ)^{jk}]−h_{ν}^{i}∂_{ν}r(λ)^{jk}+h_{ν}^{j}∂_{ν}r(λ)^{ik}−h_{ν}^{j}∂_{ν}r(λ)^{ij} = 0,

we get

(1/2) ∑_{α,β} ℓ_{α}ℓ′_{β}(λ)[a_{α}, a_{β}]^{i}[b_{α}, b_{β}]^{j}+[r(λ)^{ij}, r(λ)^{ii}]−h_{ν}^{i}(∂_{ν}r(λ))^{jj}+[h_{ν}^{j}, ∂_{ν}r(λ)^{ij}] = 0.

Since r(λ) is antisymmetric, the sum (1/2) ∑_{α,β} ... is symmetric in (i, j); antisymmetrizing in (i, j), we get

[h_{ν}^{i} + h_{ν}^{j}, ∂_{ν}r(λ)^{ij}] − h_{ν}^{i}(∂_{ν}r(λ))^{jj} + h_{ν}^{j}(∂_{ν}r(λ))^{ii} + [r(λ)^{ij}, r(λ)^{ii} + r(λ)^{jj}] = 0.

All this implies that [ρ_{𝔤,𝔥}(ȳ_{i}), ρ_{𝔤,𝔥}(ȳ_{j})] = 0.

Let us check that [x̄_{i}, t̄_{jk}] = 0 is preserved (i, j, k distinct). We have [ρ_{𝔤,𝔥}(x̄_{i}), ρ_{𝔤,𝔥}(t̄_{jk})] = [(λ^{∨})^{i}, t_{𝔤}^{jk}] = 0.

Let us prove that [ȳ_{i}, t̄_{jk}] = 0 is preserved (i, j, k distinct). We have [ρ_{𝔤,𝔥}(ȳ_{i}), ρ_{𝔤,𝔥}(t̄_{jk})] = [−h_{ν}^{i} ¯∂_{ν} + ∑_{l} r(λ)^{il}, t_{𝔤}^{jk}] = [r(λ)^{ij} + r(λ)^{ik}, t_{𝔤}^{jk}] = 0 because t_{𝔤} is 𝔤-invariant. ⊓ ⊔ Proposition 42. If V_{1}, ..., V_{n} are 𝔤-modules, then (S(𝔥)[1/P] ⊗ (⊗_{i}V_{i}))^{𝔥} is a 𝔱̄_{1,n} ⋊ 𝔡-module. The 𝔱̄_{1,n}-module structure is induced by the morphism 𝔱̄_{1,n} → H_{n}(𝔤, 𝔥) of Proposition 41, so

ρ_{(V_{i})}(x̄_{i})(f(λ) ⊗ (⊗_{i}v_{i})) = (λ^{∨})^{i}(f(λ) ⊗ (⊗_{i}v_{i})),

ρ_{(V_{i})}(ȳ_{i})(f(λ) ⊗ (⊗_{i}v_{i})) = (−h_{ν}^{i}∂_{ν} + ∑_{j} r(λ)^{ij})(f(λ) ⊗ (⊗_{i}v_{i})),

ρ_{(V_{i})}(t̄_{ij})(f(λ) ⊗ (⊗_{i}v_{i})) = t_{𝔤}^{ij}(f(λ) ⊗ (⊗_{i}v_{i})),

and the 𝔡-module structure is given by

ρ_{(V_{i})}(δ_{2m})(f(λ) ⊗ (⊗_{i}v_{i})) = \frac{1}{2}(∑_{i}\{(adλ^{∨})^{2m}(e_{α}) · e_{α}\}^{i})(f(λ) ⊗ (⊗_{i}v_{i})),

ρ_{(V_{i})}(Δ_{0})(f(λ) ⊗ (⊗_{i}v_{i})) = ( − \frac{1}{2}∂_{ν}^{2} + \frac{1}{2}⟨μ(r(λ)), h_{ν}⟩∂_{ν}

+\{\frac{1}{2}ψ(λ)^{11} − \frac{1}{2}(adλ^{∨})_{|𝔫}^{−1}(μ(r(λ))_{𝔫})\}^{12...n})(f(λ) ⊗ (⊗_{i}v_{i})),

ρ_{(V_{i})}(d)(f(λ)⊗(⊗_{i}v_{i})) = \frac{1}{2}(⟨λ, h_{ν}⟩∂_{ν}+∂_{ν}⟨λ, h_{ν}⟩+⟨μ(r(λ)), λ^{∨}⟩)(f(λ)⊗(⊗_{i}v_{i})),

ρ_{(V_{i})}(X)(f(λ) ⊗ (⊗_{i}v_{i})) = (1/2)⟨λ^{∨}, λ^{∨}⟩(f(λ) ⊗ (⊗_{i}v_{i})).

Here x_{𝔫} is the projection of x ∈ 𝔤 on 𝔫 along 𝔥.

To summarize, we have a diagram

𝔱̄_{1,n} → H_{n}(𝔤, 𝔥) → End((S(𝔥)[1/P] ⊗ (⊗_{i}V_{i}))^{𝔥})
⊂↘  (1)↑ ↗
𝔱̄_{1,n} ⋊ 𝔡

As before, the restriction morphism (S(𝔤)⊗(⊗_{i}V_{i}))^{𝔤} → (S(𝔥)[1/P]⊗(⊗_{i}V_{i}))^{𝔥} extends to a 𝔱̄_{1,n} ⋊ 𝔡-modules morphism.

The action of 𝔱̄_{1,n} ⋊𝔡 factors through a morphism ρ̃_{𝔤,𝔥} : 𝔱̄_{1,n} ⋊𝔡 → H_{n}(𝔤, 𝔥) extending ρ_{𝔤,𝔥} : 𝔱̄_{1,n} → H_{n}(𝔤, 𝔥) (denoted by (1) in the diagram).

Proof. Let λ ∈ 𝔥_{reg}^{∗}. Then if V is a 𝔤-module, we have (Ô_{𝔤^{∗},λ} ⊗V)^{𝔤} = (Ô_{𝔥^{∗},λ} ⊗ V)^{𝔥} (where Ô_{X,x} is the completed local ring of a variety X at the point x). We then have a morphism 𝔱̄_{1,n} ⋊𝔡 → H_{n}(𝔤) → End((Ô_{𝔤^{∗},λ} ⊗(⊗_{i}V_{i}))^{𝔤}) for any λ ∈ 𝔤^{∗}, so when λ ∈ 𝔥_{reg}^{∗} we get a morphism 𝔱̄_{1,n}⋊𝔡 → End((Ô_{𝔥^{∗},λ}⊗(⊗_{i}V_{i}))^{𝔥}).

Let show that the images of the generators of 𝔱̄_{1,n} ⋊𝔡 under this morphism are given by the above formulas.

Since the actions of x̄_{i}, t̄_{ij} and X on (Ô_{𝔤^{∗},λ} ⊗ (⊗_{i}V_{i}))^{𝔤} are given by multiplication by elements of (Ô_{𝔤^{∗},λ} ⊗ U(𝔤)^{⊗n})^{𝔤}, their actions on (Ô_{𝔥^{∗},λ} ⊗ (⊗_{i}V_{i}))^{𝔥} are given by multiplication by restrictions of these elements to 𝔥^{∗}.

Let us compute the action of ȳ_{i}. Let f̃(λ) ∈ (Ô_{𝔥^{∗},λ} ⊗ (⊗_{i}V_{i}))^{𝔥} and F̃(λ) ∈ (Ô_{𝔤^{∗},λ} ⊗ (⊗_{i}V_{i}))^{𝔤} be its equivariant extension to a formal map 𝔤^{∗} → ⊗_{i}V_{i}. Then for x ∈ 𝔫, we have (∂_{x^{∧}} + ∑_{i}(adλ^{∨})^{−1}(x)^{i})(F̃(λ))_{|𝔥^{∗}} = 0 (the map

x ↦ x^{∧} is the inverse of 𝔤^{∗} → 𝔤, λ ↦ λ^{∨}). Then ρ_{(V_{i})}(ȳ_{i})(f̃(λ)) = ( − h_{ν}^{i}∂_{ν} +

∑_{j} e_{β}^{i}((adλ^{∨})^{−1}(e_{β}))^{j})f̃(λ) = (−h_{ν}^{i}∂_{ν} + ∑_{j} r(λ)^{ij})(f̃(λ)).

Let us now compute the action of Δ_{0}. Let λ_{0} ∈ 𝔥^{∗} be such that λ_{0}^{∨} ∈ U and λ ∈ 𝔤^{∗} be close to λ_{0}. We set δλ := λ − λ_{0}. We then have λ = e^{adx}(λ_{0} + h^{∧}), where x ∈ 𝔫 and h ∈ 𝔥 are close to 0. We have the expansions

h = (δλ)_{𝔥}^{∨} + \frac{1}{2}[(adλ_{0}^{∨})_{|𝔫}^{−1}((δλ)_{𝔫}^{∨}), (δλ)_{𝔫}^{∨}]_{𝔥},

x = −(adλ_{0}^{∨})_{|𝔫}^{−1}((δλ)_{𝔫}^{∨}+[(adλ_{0}^{∨})_{|𝔫}^{−1}((δλ)_{𝔫}^{∨}), (δλ)_{𝔥}^{∨}]+\frac{1}{2}[(adλ_{0}^{∨})_{|𝔫}^{−1}((δλ)_{𝔫}^{∨}), (δλ)_{𝔫}^{∨}]_{𝔫})

up to terms of order > 2; here the indices u_{𝔫} and u_{𝔥} mean the projections of u ∈ 𝔤 to 𝔫 and 𝔥. If now f̃(λ) : 𝔥^{∗} ⊃ V (λ_{0}, 𝔥^{∗}) → ⊗_{i}V_{i} is an 𝔥-equivariant function defined at the vicinity of λ_{0} and F̃(λ) : 𝔤^{∗} ⊃ V (λ_{0}, 𝔤^{∗}) → ⊗_{i}V_{i} it its 𝔤-equivariant extension to a neighborhood of λ_{0} in 𝔤^{∗}, then F̃(λ) = (e^{x})^{1...n}f̃(λ_{0} + h), which implies the expansion

F̃(λ) = f̃(λ_{0}) + ((δλ)_{ν} + \frac{1}{2}⟨[(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}), e_{β′}], h_{ν}⟩(δλ)_{β}(δλ)_{β′})∂_{ν}f̃(λ_{0})

+\frac{1}{2}(δλ)_{ν}(δλ)_{ν′}∂_{νν′}^{2} f̃(λ_{0}) + ( − (adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β})(δλ)_{β}

−(adλ_{0}^{∨})^{−1}([(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}), h_{ν}])(δλ)_{ν}(δλ)_{β}

−\frac{1}{2}(adλ_{0}^{∨})_{|𝔫}^{−1}([(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}), e_{β′}]_{𝔫})(δλ)_{β}(δλ)_{β′}

+\frac{1}{2}(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β})(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β′})(δλ)_{β}(δλ)_{β′})^{1...n}f̃(λ_{0})

−(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β})^{1...n}(δλ)_{β}(δλ)_{ν}∂_{ν}f̃(λ_{0})

up to terms of order > 2.

Then

(∂_{α}^{2}F)(λ_{0}) = (∂_{ν}^{2}f̃)(λ_{0}) + ⟨[(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}], h_{ν}⟩∂_{ν}f̃(λ_{0})
+( − (adλ_{0}^{∨})_{|𝔫}^{−1}([(adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}]_{𝔫}) + ((adλ_{0}^{∨})_{|𝔫}^{−1}(e_{β}))^{2})^{1...n}f̃(λ_{0}),

which implies the formula for the action of Δ_{0}.

Then (S(𝔥)[1/P] ⊗ (⊗_{i}V_{i}))^{𝔥} ⊂ ∏_{λ∈𝔥_{reg}^{∗}}(Ô_{𝔥^{∗},λ} ⊗ (⊗_{i}V_{i}))^{𝔥} is preserved by the action of the generators of 𝔱̄_{1,n} ⋊ 𝔡-module, hence it is a sub-(𝔱̄_{1,n} ⋊ 𝔡)- module, with action given by the above formulas. ⊓ ⊔ 6.4 Realization of the universal KZB system

The realization of the flat connection d − ∑_{i} K̄_{i}(z|τ)dz_{i} − Δ̄(z|τ)dτ on (ℌ × ℂ^{n}) − Diag_{n} is a flat connection on the trivial bundle with fiber (O_{𝔥_{reg}^{∗}} ⊗ (⊗_{i}V_{i}))^{𝔥}.

We now compute this realization, under the assumption that 𝔥 ⊂ 𝔤 is a maximal abelian subalgebra. In this case, two simplifications occur:

(a) (adλ^{∨})(h_{ν}) = 0 since 𝔥 is abelian,

(b) [(adλ^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}]_{𝔫} = 0 since [(adλ^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}] commutes with any element in 𝔥, so that it belongs to 𝔥.

The image of K̄_{i}(z|τ) is then the operator

K_{i}^{(V_{i})}(z|τ) = h_{ν}^{i}∂_{ν} − ∑_{j} r(λ)^{ij} + ∑_{j|j≠i} k(z_{ij}, (adλ^{∨})^{i}|τ)(t_{𝔫}^{ij} + t_{𝔥}^{ij})

= h_{ν}^{i}∂_{ν} − r(λ)^{ii} + ∑_{j|j≠i} \frac{θ(z_{ij} + (adλ^{∨})^{i}|τ)}{θ(z_{ij}|τ)θ((adλ^{∨})^{i}|τ)}(t_{𝔫}^{ij}) + ∑_{j|j≠i} \frac{θ′}{θ} (z_{ij}|τ)t_{𝔥}^{ij}

The image of 2πiΔ̄(z|τ) is the operator

2πiΔ^{(V_{i})}(z|τ) = \frac{1}{2}∂_{ν}^{2} + \frac{1}{2}⟨[(adλ^{∨})^{−1}(e_{β}), e_{β}], h_{ν}⟩∂_{ν} − g(0, 0|τ) ∑_{i} \frac{1}{2}t_{𝔤}^{ii}

+ ∑_{i,j} \frac{1}{2}([g(z_{ij}, adλ^{∨}|τ) − (adλ^{∨})^{−2}](e_{β}))^{i}e_{β}^{j} + ∑_{i,j} \frac{1}{2}g(z_{ij}, 0|τ)h_{ν}^{i}h_{ν}^{j}

and the connection is now

∇^{(V_{i})} = d − ∑_{i} K_{i}^{(V_{i})}(z|τ)dz_{i} − Δ^{(V_{i})}(z|τ)dτ.

Recall that P (λ) = det((adλ^{∨})_{|𝔫}). We compute the conjugation P^{1/2}∇^{(V_{i})}P^{−1/2}, where P^{±1/2} is the operator of multiplication by (inverse branches of) P^{±1/2} on O_{𝔥_{reg}^{∗}} ⊗ (⊗_{i}V_{i})^{𝔥}.

Lemma 43. ∂_{ν}logP(λ) = −⟨h_{ν}, μ(r(λ))⟩, P^{1/2}[h_{ν}^{i}∂_{ν} − r(λ)^{ii}]P^{−1/2} = h_{ν}^{i}∂_{ν}, P^{1/2}[∂_{ν}^{2}+⟨[(adλ^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}], h_{ν}⟩∂_{ν}]P^{−1/2} = ∂_{ν}^{2}+∂_{ν}(⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩)−⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩^{2}.

Proof. ∂_{ν}logP(λ) = (d/dt)_{|t=0}det[(ad(λ^{∨} + th_{ν})_{|𝔫})(adλ^{∨})_{|𝔫}^{−1}] = tr[(adh_{ν})_{|𝔫} ∘ (adλ^{∨})_{|𝔫}^{−1}] = ⟨e_{β}, (adh_{ν})∘(adλ^{∨})_{|𝔫}^{−1}(e_{β})⟩ = ⟨[(adλ^{∨})_{|𝔫}^{−1}(e_{β}), e_{β}], h_{ν}⟩ = −⟨h_{ν}, μ(r(λ))⟩. The next equality follows from μ(r(λ))^{i} = 2r(λ)^{ii}. The last equality is a direct consequence. ⊓ ⊔

We then get:

Proposition 44. P^{1/2}∇^{(V_{i})}P^{−1/2} = d − ∑_{i} K̃_{i}(z|τ)dz_{i} − Δ̃(z|τ)dτ, where

K̃_{i}(z|τ) = h_{ν}^{i}∂_{ν} + ∑_{j|j≠i} \frac{θ(z_{ij} + (adλ^{∨})^{i}|τ)}{θ(z_{ij}|τ)θ((adλ^{∨})^{i}|τ)}(t_{𝔫}^{ij}) + ∑_{j|j≠i} \frac{θ′}{θ} (z_{ij}|τ)t_{𝔥}^{ij}

2πiΔ̃(z|τ) = \frac{1}{2}∂_{ν}^{2} + ∂_{ν}(⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩) − ⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩^{2} − g(0, 0|τ) ∑_{i} \frac{1}{2}t_{𝔤}^{ii}

+ ∑_{i,j} \frac{1}{2}((g(z_{ij}, adλ^{∨}|τ) − (adλ^{∨})^{−2})(e_{β}))^{i}e_{β}^{j} + ∑_{i,j} \frac{1}{2}g(z_{ij}, 0|τ)h_{ν}^{i}h_{ν}^{j},

where

g(z, 0|τ) = \frac{1}{2} \frac{θ′′}{θ} (z|τ) − 2πi\frac{∂_{τ} η}{η} (τ)

and

g(z, α|τ) − α^{−2} = \frac{1}{2} \frac{θ(z + α|τ)}{θ(x|τ)θ(α|τ)}(\frac{θ′}{θ} (z + α|τ) − \frac{θ′}{θ} (α|τ))

The term in ∑_{i}(1/2)t_{𝔤}^{ii} is central and can be absorbed by a suitable further conjugation. Rescaling t_{𝔤} into κ^{−1}t_{𝔤}, where κ ∈ ℂ^{×}, K̃_{i}(z|τ) and Δ̃(z|τ) get multiplied by κ. Moreover, we have:

Lemma 45. When 𝔤 is simple and 𝔥 ⊂ 𝔤 is the Cartan subalgebra,

∂_{ν}\{⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩\} = ⟨h_{ν}, \frac{1}{2}μ(r(λ))⟩^{2}.

Proof. Let D(λ) := ∏_{α∈Δ^{+}}(α, λ), where Δ^{+} is the set of positive roots of 𝔤. Then D(λ) is W -antiinvariant, where W is the Weyl group. Therefore ∂_{ν}^{2}D(λ) is also W -antiinvariant, so it is divisible (as a polynomial on 𝔥^{∗}) by all the (α, λ), where α ∈ Δ^{+}, so it is divisible by D(λ); since ∂_{ν}^{2}D(λ) has degree strictly lower than D(λ), we get ∂_{ν}^{2}D(λ) = 0.

Now if (e_{α}, f_{α}, h_{α}) is a basis of the 𝔰l_{2}-triple associated with α, we have r(λ) = ∑_{α∈Δ^{+}} −(e_{α}⊗f_{α}−f_{α}⊗e_{α})/(α, λ), so \frac{1}{2}μ(r(λ)) = − ∑_{α∈Δ^{+}} h_{α}/(α, λ). Therefore \frac{1}{2}μ(r(λ)) = −∂_{ν}logD(λ)h_{ν}. Then ∂_{ν}^{2}D(λ) = 0 implies that ∂_{ν}^{2}logD+ (∂_{ν}logD)^{2} = 0, which implies the lemma. ⊓ ⊔

The resulting flat connection then coincides with that of [Ber98a, FW96].

7 The universal KZB connection and representations of Cherednik algebras

7.1 The rational Cherednik algebra of type A_{n−1}

Let k be a complex number, and n ≥ 1 an integer. The rational Cherednik algebra H_{n}(k) of type A_{n−1} is the quotient of the algebra ℂ[S_{n}] ⋉ ℂ[x_{1}, ..., x_{n}, y_{1}, ..., y_{n}] by the relations

∑_{i} x_{i} = 0, ∑_{i} y_{i} = 0, [x_{i}, x_{j}] = 0 = [y_{i}, y_{j}],

[x_{i}, y_{j}] = \frac{1}{n} − ks_{ij}, i ≠ j,

where s_{ij} ∈ S_{n} is the permutation of i and j (see e.g. [EG02]). ^{8}

Let e := \frac{1}{n!} ∑_{σ∈S_{n}} σ ∈ ℂ[S_{n}] be the Young symmetrizer. The spherical subalgebra B_{n}(k) (often called the spherical Cherednik algebra) is defined to be the algebra eH_{n}(k)e.

We define an important element

h := \frac{1}{2} ∑_{i}(x_{i}y_{i} + y_{i}x_{i}).

We recall that category O is the category of H_{n}(k)-modules which are locally nilpotent under the action of the operators y_{i} and decompose into a direct sum of finite dimensional generalized eigenspaces of h. Similarly, one defines category O over B_{n}(k) to be the category of B_{n}(k)-modules which are locally nilpotent under the action of ℂ[y_{1}, ..., y_{n}]^{S_{n}} and decompose into a direct sum of finite dimensional generalized eigenspaces of h.

7.2 The homorphism from 𝔱̄_{1,n} to the rational Cherednik algebra

Proposition 46. For each k, a, b ∈ ℂ, we have a homomorphism of Lie algebras ξ_{a,b} : 𝔱̄_{1,n} → H_{n}(k), defined by the formula

x̄_{i} ↦ ax_{i}, ȳ_{i} ↦ by_{i}, t̄_{ij} ↦ ab (\frac{1}{n} − ks_{ij}) .

Proof. Straightforward. ⊓ ⊔

Remark 47. Obviously, a, b can be rescaled independently, by rescaling the generators x̄_{i} and ȳ_{i} of the source algebra 𝔱̄_{1,n}. On the other hand, if we are only allowed to apply automorphisms of the target algebra H_{n}(k), then a, b can only be rescaled in such a way that the product ab is preserved.

This shows that any representation V of the rational Cherednik algebra H_{n}(k) yields a family of realizations for 𝔱̄_{1,n} parametrized by a, b ∈ ℂ, and gives rise to a family of flat connections ∇_{a,b} over the configuration space C̄(E_{τ} , n).

8The generators x_{α}, ∂_{α} of Section 6.1 will be henceforth renamed q_{α}, p_{α}.

7.3 Monodromy representations of double affine Hecke algebras

Let H_{n}(q, t) be Cherednik’s double affine Hecke algebra of type A_{n−1}. By definition, H_{n}(q, t) is the quotient of the group algebra of the orbifold fundamental group \overline{B}_{1,n} of C̄(E_{τ} , n)/S_{n} by the additional relations

(T − q^{−1}t)(T + q^{−1}t^{−1}) = 0,

where T is any element of \overline{B}_{1,n} homotopic (as a free loop) to a small loop around the divisor of diagonals in the counterclockwise direction.

Let V be a representation of H_{n}(k), and let ∇_{a,b}(V) be the universal connection ∇_{a,b} evaluated in V. In some cases, for example if a, b are formal, or if V is finite dimensional, we can consider the monodromy of this connection, which obviously gives a representation of H_{n}(q, t) on V, with

q = e^{−2πiab/n}, t = e^{−2πikab}.

In particular, taking a = b, V = H_{n}(k), this monodromy representation defines an homomorphism θ_{a} : H_{n}(q, t) → H_{n}(k)[[a]], where

q = e^{−2πia^{2}/n}, t = e^{−2πika^{2}}.

It is easy to check that this homomorphism becomes an isomorphism upon inverting a. The existence of such an isomorphism was pointed out by Cherednik (see [Che03], end of Section 6, and the end of [Che97]), but his proof is different.

Example 48. Let k = r/n, where r is an integer relatively prime to n. In this case, it is known (see e.g. [BEG03a]) that the algebra H_{n}(k) admits an irreducible finite dimensional representation Y (r, n) of dimension r^{n−1}. By virtue of the above construction, the space Y (r, n) carries an action of H_{n}(q, t) with any nonzero q, t such that q^{r} = t. This finite dimensional representation of H_{n}(q, t) is irreducible for generic q, and is called a perfect representation; it was first constructed in [Eti94], p. 500, and later in [Che03], Theorem 6.5, in a greater generality.

7.4 The modular extension of ξ_{a,b}.

Assume that a, b ≠ 0.

Proposition 49. The homomorphism ξ_{a,b} can be extended to the algebra U(𝔱̄_{1,n} ⋊ 𝔡) ⋊ S_{n} by the formulas

ξ_{a,b}(s_{ij}) = s_{ij},

ξ_{a,b}(d) = h = \frac{1}{2} ∑_{i}(x_{i}y_{i} + y_{i}x_{i}), ξ_{a,b}(X) = −\frac{1}{2}ab^{−1} ∑_{i} x_{i}^{2},

ξ_{a,b}(Δ_{0}) = \frac{1}{2}ba^{−1} ∑_{i} y_{i}^{2}, ξ_{a,b}(δ_{2m}) = −\frac{1}{2}a^{2m−1}b^{−1} ∑_{i<j}(x_{i} − x_{j})^{2m}.

Proof. Direct computation. ⊓ ⊔

Thus, the flat connections ∇_{a,b} extend to flat connections on M_{1,[n]}.

This shows that the monodromy representation of the connection ∇_{a,b}(V), when it can be defined, is a representation of the double affine Hecke algebra H_{n}(q, t) with a compatible action of the extended modular group { \atop SL_{2}(ℤ)}. In particular, this is the case if V = Y (r, n). Such representations of { \atop SL_{2}(ℤ)} were considered by Cherednik, [Che03]. The element T of { \atop SL_{2}(ℤ)} acts in this representation by “the Gaussian”, and the element S by the “Fourier-Cherednik transform”. They are generalizations of the { \atop SL_{2}(ℤ)}-action on Verlinde algebras.

8 Explicit realizations of certain highest weight representations of the rational Cherednik algebra of type A_{n−1}

8.1 The representation V_{N}.

Let N be a divisor of n, and 𝔤 = 𝔰𝔩_{N}(ℂ), G = SL_{N}(ℂ). Let V_{N} = (ℂ[𝔤] ⊗ (ℂ^{N})^{⊗n})^{𝔤} (the divisor condition is needed for this space to be nonzero). It turns out that V_{N} has a natural structure of a representation of H_{n}(k) for k = N/n.

Proposition 50. We have a homomorphism ζ_{N} : H_{n}(N/n) → End(V_{N}), defined by the formulas

ζ_{N}(s_{ij}) = s_{ij}, ζ_{N}(x_{i}) = X_{i}, ζ_{N}(y_{i}) = Y_{i}, (i = 1, ..., n)

where for f ∈ V_{N}, A ∈ 𝔤 we have

(X_{i}f)(A) = A_{i}f(A),

(Y_{i}f)(A) = \frac{N}{n} ∑_{p}(b_{p})_{i} \frac{∂f}{∂b_{p}} (A),

where {b_{p}} is an orthonormal basis of 𝔤 with respect to the trace form.

Proof. Straightforward verification. ⊓ ⊔

The relationship of the representation V_{N} to other results in this paper is described by the following proposition.

Proposition 51. The connection ∇_{a,1}(V_{N}) corresponding to the representation V_{N} is the usual KZB connection for the n-point correlation functions on the elliptic curve for the Lie algebra 𝔰𝔩_{N} and n copies of the vector representation ℂ^{N}, at level K = −\frac{n}{aN} − N.

Proof. We have a sequence of maps

U(𝔱̄_{1,n} ⋊ 𝔡) ⋊ S_{n} → H_{n}(N/n) → H_{n}(𝔤) ⋊ S_{n} → End(V_{N}),

where the first map is ξ_{a,b}, the second map sends s_{ij} to s_{ij}, x_{i} to the class of ∑_{α} q_{α} ⊗e_{α}^{i}, and y_{i} to the class of ∑_{α} p_{α} ⊗e_{α}^{i} (recall that the x_{a}, ∂_{a} of Section 6.1 have been renamed q_{a}, p_{a}), and the last map is explained in Section 6.1. The composition of the two first maps is then that of Proposition 40, and the composition of the two last maps is the map ζ_{N} of Proposition 50. This implies the statement. ⊓ ⊔

Remark 52. Suppose that K is a nonnegative integer, i.e. a = −\frac{n}{N(K+N)}, where K ∈ ℤ_{+}. Then the connection ∇_{a,1} on the infinite dimensional vector bundle with fiber V_{N} preserves a finite dimensional subbundle of conformal blocks for the WZW model at level K. Th subbundle gives rise to a finite dimensional monodromy representation V_{N}^{K} of the Cherednik algebra H_{n}(q, t) with

q = e^{\frac{2πi}{N(K+N)}} , t = q^{N},

(so both parameters are roots of unity). The dimension of V_{N}^{K} is given by the

Verlinde formula, and it carries a compatible action of { \atop SL_{2}(ℤ)} to the action of the Cherednik algebra. Representations of this type were studied by Cherednik in [Che03].

8.2 The spherical part of V_{N}.

Note that

((∑_{i=1}^{n} X_{i}^{p})f)(A) = \frac{n}{N} (tr A^{p})f(A),  (43)

((∑_{i=1}^{n} Y_{i}^{p})f)(A) = (\frac{N}{n} )^{p−1} (tr ∂_{A}^{p})f(A)  (44)

Consider the space U_{N} = eV_{N} = (ℂ[𝔤] ⊗ S^{n}ℂ^{N})^{𝔤} as a module over the spherical subalgebra B_{n}(k). It is known (see e.g. [BEG03b]) that the spherical subalgebra is generated by the elements (∑ x_{i}^{p})e and (∑ y_{i}^{p})e. Thus formulas (43,44) determine the action of B_{n}(k) on U_{N}.

We note that by restriction to the set 𝔥 of diagonal matrices diag(λ_{1}, ..., λ_{N}), and dividing by Δ^{n/N}, where Δ = ∏_{i<j}(λ_{i} − λ_{j}), one identifies U_{N} with ℂ[𝔥]^{S_{N}}. Moreover, it follows from [EG02] that formulas (43,44) can be viewed as defining an action of another spherical Cherednik algebra, namely B_{N}(1/k), on ℂ[𝔥]^{S_{N}}. Moreover, this representation is the symmetric part W of the standard polynomial representation of H_{N}(1/k), which is faithful and irreducible since 1/k = n/N is an integer ([GGOR03]). In other words, we have the following proposition.

Proposition 53. There exists a surjective homomorphism ϕ : B_{n}(N/n) → B_{N}(n/N), such that ϕ^{∗}W = U_{N}. In particular, U_{N} is an irreducible representation of B_{n}(N/n).

Proposition 53 can be generalized as follows. Let 0 ≤ p ≤ n/N be an integer. Consider the partition μ(p) = (n − p(N − 1), p, ..., p) of n. The representation of 𝔤 attached to μ(p) is S^{n−pN}ℂ^{N}.

Let e(p) be a primitive idempotent of the representation of S_{n} attached to μ(p). Let U_{N}^{p} = e(p)V_{N} = (ℂ[𝔤]⊗S^{n−pN}ℂ^{N})^{𝔤}. Then the algebra e(p)H_{n}(N/n)e(p) acts on U_{N}^{p}, and the above situation of U_{N} is the special case p = 0.

Proposition 54. There exists a surjective momorphism ϕ_{p} : e(p)H_{n}(N/n)e(p) → B_{N}(n/N − p), such that ϕ_{p}^{∗}W = U_{N}^{p}. In particular, U_{N}^{p} is an irreducible representation of B_{n}(N/n − p).

Proof. Similar to the proof of Proposition 53. ⊓ ⊔

Example 55. p = 1, n = N. In this case e(p) = e_{−} = \frac{1}{n!} ∑_{σ∈S_{n}} ε(σ)σ, the antisymmetrizer, and the map ϕ_{p} is the shift isomorphism e_{−}H_{N}(1)e_{−} → eH_{N}(0)e.

8.3 Coincidence of the two 𝔰𝔩_{2} actions

As before, let {b_{p}} be an orthonormal basis of 𝔤 (under some invariant inner product). Consider the 𝔰𝔩_{2}-triple

H = ∑ b_{p} \frac{∂}{∂b_{p}} + \frac{dim 𝔤}{2}  (45)

(the shifted Euler field),

F = \frac{1}{2} ∑_{p} b_{p}^{2}, E = \frac{1}{2}Δ_{𝔤},  (46)

where Δ_{𝔤} is the Laplace operator on 𝔤. Recall also (see e.g. [BEG03b]) that the rational Cherednik algebra contains the 𝔰𝔩_{2}-triple h = \frac{1}{2} ∑_{i}(x_{i}y_{i} + y_{i}x_{i}), e = \frac{1}{2} ∑_{i} y_{i}^{2}, f = \frac{1}{2} ∑_{i} x_{i}^{2}.

The following proposition shows that the actions of these two 𝔰𝔩_{2} algebras on V_{N} essentially coincide.

Proposition 56. On V_{N}, one has

h = H, e = \frac{N}{n} E, f = \frac{n}{N} F.

Proof. The last two equations follow from formulas (43,44), and the first one follows from the last two by taking commutators. ⊓ ⊔ 8.4 The irreducibility of V_{N}.

Let Δ(n, N) be the representation of the symmetric group S_{n} corresponding to the rectangular Young diagram with N rows (and correspondingly n/N columns), i.e. to the partition (\frac{n}{N} , ..., \frac{n}{N}); e.g., Δ(n, 1) is the trivial representation.

For a representation π of S_{n}, let L(π) denote the irreducible lowest weight representation of H_{n}(k) with lowest weight π.

Theorem 57. The representation V_{N} is isomorphic to L(Δ(n, N)).

Proof. The representation V_{N} is graded by the degree of polynomials, and in degree zero we have V_{N}[0] = ((ℂ^{N})^{⊗n})^{𝔤} = Δ(n, N) by the Weyl duality.

Let us show that the module V_{N} is semisimple. It is sufficient to show that V_{N} is a unitary representation, i.e. admits a positive definite contravariant Hermitian form. Such a form can be defined by the formula

(f, g) = ⟨f(∂_{A}), g(A)⟩|_{A=0},

where ⟨−, −⟩ is the Hermitian form on (ℂ^{N})^{⊗n} obtained by tensoring the standard forms on the factors. This form is obviously positive definite, and satisfies the contravariance properties:

(Y_{i}f, g) = \frac{N}{n} (f, X_{i}g), (f, Y_{i}g) = \frac{N}{n} (X_{i}f, g).

The existence of the form (−, −) implies the semisimplicity of V_{N}. In particular, we have a natural inclusion L(Δ(n, N)) ⊂ V_{N}.

Next, formula (43) implies that V_{N} is a torsion-free module over R := ℂ[x_{1}, ..., x_{N}]^{S_{N}} = ℂ[∑_{i=1}^{N} x_{i}^{p}, 2 ≤ p ≤ N]. Since V_{N} is semisimple, this implies that V_{N}/L(Δ(n, N)) is torsion-free as well.

On the other hand, we will now show that the quotient V_{N}/L(Δ(n, N)) is a torsion module over R. This will imply that the quotient is zero, as desired.

Let v_{1}, ..., v_{N} be the standard basis of ℂ^{N}, and for each sequence J = (j_{1}, ..., j_{n}), j_{i} ∈ {1, ..., N}, let v_{J} := v_{j_{1}} ⊗ ... ⊗ v_{j_{n}}. Let us say that a sequence J is balanced if it contains each of its members exactly n/N times. Let B be the set of balanced sequences. The set B has commuting left and right actions S_{N} and S_{n}, σ ∗ (j_{1}, ..., j_{n}) ∗ τ = (σ(j_{τ(1)}), ..., σ(j_{τ(n)})). Let J_{0} = (1...1, 2...2, ..., N...N), then any J ∈ B has the form J = J_{0} ∗ τ for some τ ∈ S_{n}.

Let f ∈ V_{N}. Then f is a function 𝔥 → ((ℂ^{N})^{⊗n})^{𝔥}, equivariant under the action of S_{N} (here 𝔥 ⊂ 𝔤 is the Cartan subalgebra, so 𝔥 = {(λ_{1}, ..., λ_{N})| ∑_{i} λ_{i} = 0}), so

f(λ) = ∑_{J∈B} f_{J}(λ)v_{J},  (47)

where λ = (λ_{1}, ..., λ_{N}), and f_{J} are scalar functions (the summation is over B since f(λ) must have zero weight). By the S_{N}-invariance, we have f_{σ∗J}(σ(λ)) = f_{J}(λ). We then decompose f(λ) = ∑_{o∈S_{N} \B} f_{o}(λ), where f_{o}(λ) = ∑_{J∈o} f_{J}(λ)v_{J}.

For each o ∈ S_{N} \ B, we construct a nonzero ϕ_{o} ∈ ℂ[x_{1}, ...., x_{n}] such that ϕ_{o} · f_{o}(λ) ∈ L(Δ(n, N)). Then ϕ := ∏_{o∈S_{N} \B} ∏_{σ∈S_{N}} σ(ϕ_{o}) ∈ R is nonzero and such that ϕ · f(λ) ∈ L(Δ(n, N)).

We first construct ϕ_{o} when o = o_{0}, the class of J_{0}. By S_{N}-invariance, f_{o_{0}}(λ) has the form

f_{o_{0}}(λ) = ∑_{σ∈S_{N}} g(λ_{σ(1)}, ..., λ_{σ(N)})v_{σ(1)}^{⊗n/N} ⊗ ... ⊗ v_{σ(N)}^{⊗n/N}

where g(λ, ..., λ_{N}) ∈ ℂ[λ_{1}, ..., λ_{N}]. For ϕ_{o_{0}} ∈ ℂ[x_{1}, ..., x_{N}], we have

ϕ_{o_{0}} · f_{o_{0}}(λ) = ∑_{σ∈S_{N}}(ϕ_{o_{0}}g)(λ_{σ(1)}, ..., λ_{σ(N)})v_{σ(1)}^{⊗n/N} ⊗ ... ⊗ v_{σ(N)}^{⊗n/N}.  (48)

On the other hand, let v ∈ Δ(n, N); expand v = ∑_{J∈B} c_{J}v_{J}. One checks that v can be chosen such that c_{J_{0}} ≠ 0 (one starts with a nonzero vector v′ and J′ ∈ B such that the coordinate of v′ along J′ is nonzero, and then acts on v′ by an element of S_{n} bringing J′ to J_{0}). Then since v is 𝔤-invariant (and therefore S_{N}-invariant), we have

c_{σ(1)...σ(1)...σ(N)...σ(N)} = c_{J_{0}}  (49)

for any σ ∈ S_{N}.

If Q ∈ ℂ[x_{1}, ..., x_{n}], then

(Q·v)(λ) = ∑_{(j_{1},...,j_{n})∈B} c_{j_{1}...j_{n}}Q(λ_{j_{1}}, ..., λ_{j_{n}})v_{j_{1}} ⊗...⊗v_{j_{n}} ∈ L(Δ(n, N)). (50)

Set Q_{0}(λ_{1}, ..., λ_{n}) := ∏_{1≤a<b≤n,j_{a}^{0}≠j_{b}^{0}}(λ_{a} − λ_{b}), where (j_{1}^{0}, ..., j_{n}^{0}) = J_{0}, q_{0}(λ_{1}, ..., λ_{N}) := Q_{0}(λ_{1}...λ_{1}, ..., λ_{N}...λ_{N}), so q_{0}(λ_{1}, ..., λ_{N}) = ( ∏_{1≤i<j≤N}(λ_{i}−

λ_{j}))^{(n/N)^{2}}.

Set ϕ_{o_{0}}(λ_{1}, ..., λ_{N}) := q_{0}(λ_{1}, ...., λ_{N}) and

Q(λ_{1}, ..., λ_{n}) := Q_{0}(λ_{1}, ..., λ_{n})q(λ_{1}, λ_{(n/N)+1}, ..., λ_{(N−1) \frac{n}{N} +1}).

Then (48) and (50) coincide, as: (a) for J ∉ o_{0}, Q_{0}(λ_{j_{1}}, ..., λ_{j_{n}}) = 0 so the coefficient of v_{J} in both expressions is zero, (b) the coefficients of v_{J_{0}} in both expressions coincide, (c) for J ∈ o_{0}, the coefficients of v_{J} coincide because of (b) and of (49). The functions ϕ_{o} are constructed in the same way for a general o ∈ S_{N} \ B. This ends the proof of the theorem. ⊓ ⊔

Remark 58. Theorem 57 is a special case of a much more general (but much less elementary) Theorem 68, which is proved below.

8.5 The character formula for V_{N}.

For each partition μ of n, let V (μ) be the representation of 𝔤, and π(μ) the representation of S_{n} corresponding to μ.

Let P_{μ}(q) be the q-analogue of the weight multiplicity of the zero weight in V (μ). Namely, we have a filtration F^{•} on V (μ)[0] such that F^{i} is the space of vectors in V (μ)[0] killed by the i + 1-th power of the principal nilpotent element ∑ e_{i} of 𝔤. Then P_{μ}(q) = ∑_{j≥0} dim(F^{j}/F^{j−1})q^{j}. The coefficients of P_{μ}(q) are called the generalized exponents of V (μ) (see [Kos63, Hes80, Lus81] for more details).

We have V_{N} = ⊕_{μ}π(μ) ⊗ (ℂ[𝔤] ⊗ V (μ))^{𝔤}. This together with Theorem 57 implies the following.

Corollary 59. The character of L(Δ(n, N)) is given by the formula

Tr|_{L(Δ(n,N))}(w · q^{h}) = q^{(N^{2}−1)/2} \frac{∑_{μ} χ_{π(μ)}(w)P_{μ}(q)}{(1 − q^{2})...(1 − q^{N})},

where w ∈ S_{n}, and χ_{π(μ)} is the character of π(μ). Here the summation is over partitions μ of n with at most N parts.

Proof. The formula follows, using Proposition 56, from Kostant’s result [Kos63] that (ℂ[𝔤] ⊗ V (μ))^{𝔤} is a free module over ℂ[𝔤]^{𝔤}, and the fact that the Hilbert polynomial of the space of generators for this module is the q-weight multiplicity of the zero weight, P_{μ}(q) [Kos63, Lus81, Hes80]. ⊓ ⊔

Remark 60. It would be interesting to compare this formula with the character formula of [Rou05] for the same module.

9 Equivariant D-modules and representations of the rational Cherednik algebra

9.1 The category of equivariant D-modules on the nilpotent cone

The theory of equivariant D-modules on the nilpotent cone arose from Harish-Chandra’s work on invariant distributions on nilpotent orbits of real groups, and was developed further in many papers, see e.g. [HK84, LS97, Lev98, Mir04] and references therein. Let us recall some of the basics of this theory.

Let G be a simply connected simple algebraic group over ℂ, and 𝔤 its Lie algebra. Let N ⊂ 𝔤 be the nilpotent cone of 𝔤. We denote by D(𝔤) the category of finitely generated D-modules on 𝔤, by D_{G}(𝔤) the subcategory of G-equivariant D-modules, and by D_{G}(N) the category of G-equivariant Dmodules which are set-theoretically supported on N (here we do not make a distinction between a D-module on an affine space and the space of its global sections). Since G acts on N with finitely many orbits, it is well known that any object in D_{G}(N) is regular and holonomic.

Moreover, the category D_{G}(N) has finitely many simple objects, and every object of this category has finite length (so this category is equivalent to the category of modules over a finite dimensional algebra).

9.2 Simple objects in D_{G}(N)

Recall (see e.g. [Mir04] and references) that irreducible objects in the category D_{G}(N) are parametrized by pairs (O, χ), where O is a nilpotent orbit of G in 𝔤, and χ is an irreducible representation of the fundamental group π_{1}(O), which is clearly isomorphic to the component group A(O) of the centralizer G_{x} of a point x ∈ O. Namely, χ defines a local system L_{χ} on O, and the simple object M(O, χ) ∈ D_{G}(N) is the direct image of the Goresky-Macpherson extension of L_{χ} to the closure Ō of O, under the inclusion of Ō into 𝔤.

9.3 Semisimplicity of D_{G}(N).

The proof of the following theorem was explained to us by G. Lusztig.

Theorem 61. The category D_{G}(N) is semisimple.

Proof. We may replace the category D_{G}(N) by the category of G-equivariant perverse sheaves (of complex vector spaces) on 𝔤 supported on N, Perv_{G}(N), as these two categories are known to be equivalent. We must show that Ext^{1}(P, Q) = 0 for every two simple objects P, Q ∈ Perv_{G}(N).

Let P′, Q′ be the Fourier transforms of P, Q. Then P′, Q′ are character sheaves on 𝔤, and it suffices to show that Ext^{1}(P′, Q′) = 0.

Recall that to each character sheaf S one can naturally attach a conjugacy class of pairs (L, θ), where L is a Levi subgroup of G, and θ is a cuspidal local system on a nilpotent orbit for L. It is shown by arguments parallel to those in [Lus85] (which treats the more difficult case of character sheaves on the group) that if (L_{i}, θ_{i}) corresponds to S_{i}, i = 1, 2, and (L_{1}, θ_{1}) is not conjugate to (L_{2}, θ_{2}) then Ext^{∗}(S_{1}, S_{2}) = 0. Thus it is sufficient to assume that the pair (L, θ) attached to P′ and Q′ is the same.

Using standard properties of constructible sheaves (in particular, Poincaré duality), we have

Ext^{1}(P′, Q′) = H^{1}(𝔤, \underline{Hom}(P′, Q′)) =
H_{c}^{2 dim 𝔤−1}(𝔤, \underline{Hom}(P′, Q′)^{∗})^{∗} = H_{c}^{2 dim 𝔤−1}(𝔤, (Q′)^{∗} ⊗ P′)^{∗},

where ∗ for sheaves denotes the Verdier duality functor.

Recall that to each character sheaf one can attach an irreducible representation of a certain Weyl group, via the generalized Springer correspondence. Let R be the direct sum of all character sheaves corresponding to a given pair (L, θ) with multiplicities given by the dimensions of the corresponding representations. Then it is sufficient to show that H_{c}^{2 dim 𝔤−1}(𝔤, (R′)^{∗}⊗R′) = 0.

This fact is essentially proved in [Lus88]. Namely, it follows from the computations of [Lus88] that H_{c}^{i}(𝔤, (R′)^{∗} ⊗ R′) is the cohomology with compact support of a certain generalized Steinberg variety with twisted coefficients, and it is shown that this cohomology is concentrated in even degrees.^{9} The theorem is proved. ⊓ ⊔

9.4 Monodromicity

We will need the following lemma.

Lemma 62. Let Q ∈ D_{G}(N). Then for any finite dimensional representation U of 𝔤, the action of the shifted Euler operator H defined by (45) on (Q⊗U)^{𝔤} is locally finite (so Q is a monodromic D-module), and has finite dimensional generalized eigenspaces. Moreover, the eigenvalues of H on (Q ⊗ U)^{𝔤} are bounded from above. In particular, (Q ⊗ U)^{𝔤} belongs to category O for the 𝔰𝔩_{2}-algebra spanned by H and the elements E, F given by (46).

Proof. Since Q has finite length, it is sufficient to assume that Q is irreducible. We may further assume that Q is generated by an irreducible G-submodule V, annihilated by multiplication by any invariant polynomial on 𝔤 of positive degree. Indeed, let V_{0} be an irreductible G-submodule of Q, let J_{V_{0}} := {f ∈ ℂ[𝔤]^{𝔤}|fV_{0} = 0} and for any v ∈ V_{0}, let J_{v} := {f ∈ ℂ[𝔤]^{𝔤}|fv = 0}. Then if v ∈ V_{0} is nonzero, J_{v} = J_{V_{0}} as Gv = V_{0}. Moreover, the support condition implies that J_{v} ⊂ 𝔪^{k} for some k ≥ 0, where 𝔪 = ℂ[𝔤]_{+}^{𝔤}. So J_{V_{0}} ⊂ 𝔪^{k} and is an ideal of ℂ[𝔤]^{𝔤}. Let f ∈ ℂ[𝔤]^{𝔤} be such that f ∉ J_{V_{0}} and f𝔪 ⊂ J_{V_{0}}; we set V := fV_{0}.

Then Q is a quotient of the D-module Q̃ ⊗ V by a G-stable submodule, where

Q̃ := D(𝔤)/(D(𝔤)ad(Ann(V)) + D(𝔤)I),

Ann(V) is the annihilator of V in U(𝔤), and I is the ideal in ℂ[𝔤] generated by invariant polynomials on 𝔤 of positive degree. Thus, it suffices to show that the lemma holds for the module Q̃ (which is only weakly G-equivariant, i.e. the group action and the Lie algebra action coming from differential operators do not agree, in general).

The algebra D(𝔤) has a grading in which deg(𝔤^{∗}) = −1, deg(𝔤) = 1. This grading descends to a grading on Q̃. We will show that for each U, this grading on (Q̃ ⊗ U)^{𝔤} has finite dimensional pieces, and is bounded from above. This implies the lemma, since the Euler operator preserves the grading.

Consider the associated graded module Q̃_{0} of Q̃ under the Bernstein filtration. This is a bigraded module over ℂ[𝔤 ⊕ 𝔤] (where we identify 𝔤 and 𝔤^{∗}

9More precisely, in the arguments of [Lus88] the vanishing of odd cohomology is proved for G-equivariant cohomology with compact supports, and in the nonequivariant case one should use parallel arguments, rather than exactly the same arguments.

using the trace form). We have to show that the homogeneous subspaces of (Q̃_{0} ⊗ U)^{𝔤} under the grading defined by deg(𝔤 ⊕ 0) = −1, deg(0 ⊕ 𝔤) = 1 are finite dimensional.

The associated graded of the ideal Ann(V) ⊂ U(𝔤) is such that ℂ[𝔤]_{+}^{k} ⊂ grAnn(V) ⊂ ℂ[𝔤]_{+} for some k ≥ 1, therefore

Q̃_{0} = ℂ[𝔤 ⊕ 𝔤]/J,

where J is a (not necessarily radical) ideal whose zero set is the variety Z of pairs (u, v) ∈ N × 𝔤 such that [u, v] = 0. Let

Q′_{0} = ℂ[𝔤 ⊕ 𝔤]/\sqrt{J}.

Because of the Hilbert basis theorem, it suffices to prove that the homogeneous subspaces of (Q′_{0}⊗U)^{𝔤} are finite dimensional, and the degree is bounded above. But Q′_{0} is the algebra of regular functions on Z. By the result of [Jos97], one has ℂ[Z]^{𝔤} = ℂ[𝔤]^{𝔤}, the algebra of invariant polynomials of Y. But it follows from the Hilbert’s theorem on invariants that every isotypic component of ℂ[Z] is a finitely generated module over ℂ[Z]^{𝔤}. This implies the result. ⊓ ⊔

9.5 Characters

Lemma 62 allows one to define the character of an object M ∈ D_{G}(N). Namely, let μ = (μ_{1}, ..., μ_{N}) be a dominant integral weight for 𝔤, and V (μ) the irreducible representation of 𝔤 with highest weight μ. Let K_{M}(μ) = (M ⊗ V (μ))^{𝔤}. Then the character of M is defined by the formula

Ch_{M}(t, g) = Tr_{|M}(gt^{−H}) = ∑_{μ} Tr_{|K_{M} (μ)}(t^{−H})χ_{μ}(g), g ∈ G,

where χ_{μ} denotes the character of μ. It can be viewed as a linear functional from ℂ[G]^{G} to 𝔽 := ⊕_{β∈ℂ}t^{β}ℂ[[t]], via the integration pairing.

In other words, the multiplicity spaces K_{M}(μ) are representations from category O of the Lie algebra 𝔰𝔩_{2} spanned by E, F, H, and the character of M carries the information about the characters of these representations.

The problem of computing characters of simple objects in D_{G}(N) is interesting and, to our knowledge, open. Below we will show how these characters for G = SL_{N}(ℂ) can be expressed via characters of irreducible representations of the rational Cherednik algebra.

Example 63. Recall (see e.g. [Mir04]) that an object M ∈ D_{G}(N) is cuspidal iff F(M) ∈ D_{G}(N), where F is the Fourier transform (Lusztig’s criterion). If follows that in the case of cuspidal objects M, the spaces K_{M}(μ) are also in the category O for the opposite Borel subalgebra of 𝔰l_{2}, hence are finite dimensional representations of 𝔰𝔩_{2}, and, in particular, their dimensions are of interest.

9.6 The functors F_{n}, F_{n}^{∗}

The representation V_{N} is a special case of representations of the rational Cherednik algebra which can be constructed via a functor similar to the one defined in [GG04]. Namely, the construction of V_{N} can be generalized as follows.

Let n and N be positive integers (we no longer assume that N is a divisor of n), and k = N/n. We again consider the special case G = SL_{N}(ℂ), 𝔤 = 𝔰𝔩_{N}(ℂ). Then we have a functor F_{n} : D(𝔤) → H_{n}(k)-mod defined by the formula

F_{n}(M) = (M ⊗ (ℂ^{N})^{⊗n})^{𝔤},

where 𝔤 acts on M by adjoint vector fields. The action of H_{n}(k) on F_{n}(M) is defined by the same formulas as in Proposition 50, and Proposition 56 remains valid.

Note that F_{n}(M) = F_{n}(M_{fin}), where M_{fin} is the set of 𝔤-finite vectors in M. Clearly M_{fin} is a G-equivariant D-module. Thus, it is sufficient to consider the restriction of F_{n} to the subcategory D_{G}(𝔤), which we will do from now on.

In general, F_{n}(M) does not belong to category O. However, we have the following lemma.

Lemma 64. If the Fourier transform F(M) of M is set-theoretically supported on the nilpotent cone N of 𝔤, then F_{n}(M) belongs to the category O.

Proof. Since F(M) is supported on N, invariant polynomials on 𝔤 act locally nilpotently on F(M). Hence invariant differential operators on 𝔤 with constant coefficients act locally nilpotently on M. Thus, it follows from formula (44) that the algebra ℂ[y_{1}, ..., y_{n}]^{S_{n}} acts locally nilpotently on F_{n}(M). Also, by Lemma 62, the operator h acts with finite dimensional generalized eigenspaces on F_{n}(M). This implies the statement. ⊓ ⊔

Thus we obtain an exact functor F_{n}^{∗} = F_{n} ∘ F : D_{G}(N) → O(H_{n}(k)).

9.7 The symmetric part of F_{n}

Consider the symmetric part eF_{n}(M) of F_{n}(M). We have eF_{n}(M) = (M ⊗ S^{n}ℂ^{N})^{𝔤}, and we have an action of the spherical subalgebra B_{n}(k) on eF_{n}(M), given by formulas (43,44).

This allows us to relate the functor F_{n} with the functor defined in [GG04]. Namely, recall from [GG04] that for any c ∈ ℤ, one may define the category D_{c}(𝔤 × ℙ^{N−1}) of coherent D-modules on 𝔤 × ℙ^{N−1} which are twisted by the c-th power of the tautological line bundle on the second factor (this makes sense for all complex c even though the c-th power is defined only for integer c). Then the paper [GG04]^{10} defines a functor

10There seems to be a misprint in [GG04]: in the definition of ℍ, c should be replaced by c/N.

ℍ : D_{c}(𝔤 × ℙ^{N−1}) → B_{N}(c/N)-mod,

given by ℌ(M) = M^{𝔤}.

Proposition 65. (i) If n is divisible by N then one has a functorial isomorphism eF_{n}(M) ≃ ϕ^{∗}ℍ(M ⊗ S^{n}ℂ^{N}), where S^{n}ℂ^{N} is regarded as a twisted D-module on ℙ^{N−1} (with c = n).

(ii) For any n, the actions of B_{n}(N/n) and B_{N}(n/N) on the space eF_{n}(M) = ℍ(M ⊗ S^{n}ℂ^{N}) have the same image in the algebra of endomorphisms of this space.

Proof. This follows from the definition of ℍ and formulas (43,44). ⊓ ⊔

Corollary 66. The functor eF_{n}^{∗} on the category D_{G}(N) maps irreducible objects into irreducible ones.

Proof. This follows from Proposition 65, (ii) and Proposition 7.4.3 of [GG04], which states that the functor ℍ maps irreducible objects to irreducible ones.

⊓ ⊔

Formulas 43,44 can also be used to study the support of F_{n}^{∗}(M) for M ∈ D_{G}(N), as a ℂ[x_{1}, ..., x_{n}]-module. Namely, we have the following proposition.

Proposition 67. Let q = GCD(n, N) be the greatest common divisor of n and N. Then the support S of F_{n}^{∗}(M) is contained in the union of the S_{n}- translates of the subspace E_{q} of ℂ^{n} defined by the equations ∑_{i=1}^{n} x_{i} = 0 and x_{i} = x_{j} if \frac{n}{q}(l − 1) + 1 ≤ i, j ≤ \frac{nl}{q} for some 1 ≤ l ≤ q.

Proof. It follows from equation (44) that for any (x_{1}, ..., x_{n}) ∈ S there exists a point (z_{1}, ..., z_{N}) ∈ ℂ^{N} such that one has

\frac{1}{n} ∑_{i=1}^{n} x_{i}^{p} = \frac{1}{N} ∑_{j=1}^{N} z_{j}^{p}

for all positive integer p. In particular, writing generating functions, we find that

N ∑_{i=1}^{n} \frac{1}{1 − tx_{i}} = n ∑_{j=1}^{N} \frac{1}{1 − tz_{j}} .

In particular, every fraction occurs on both sides at least LCM(n, N) times, and hence the numbers x_{i} fall into n/q-tuples of equal numbers (and the numbers z_{j} into N/q-tuples of equal numbers). The proposition is proved. ⊓ ⊔ 9.8 Irreducible equivariant D-modules on the nilpotent cone for G = SL_{N}(ℂ)

Nilpotent orbits for SL_{N}(ℂ) are labelled by Young diagrams, or partitions. Namely, if x ∈ 𝔰𝔩_{N}(ℂ) is a nilpotent element, then we let μ_{i} be the sizes of its Jordan blocks enumerated in the decreasing order. The partition μ = (μ_{1}, ..., μ_{m}) and the corresponding Young diagram whose rows have lengths μ_{i} are attached to x. If O is the orbit of x then we will denote μ by μ(O). For instance, if O = {0} then μ = (1^{N}) and if O is the open orbit then μ = (N).

It is known (and easy to show) that the group A(O) is naturally isomorphic to ℤ/dℤ, where d is the greatest common divisor of the μ_{i}. Namely, let Z = ℤ/Nℤ be the center of G (we identify ℤ/Nℤ with Z by p → e^{2πip/N}Id). Then we have a natural surjective homomorphism θ : Z → A(O) induced by the inclusion Z → G_{x}, x ∈ O. This homomorphism sends d to 0, and thus A(O) gets identified with ℤ/dℤ.

Thus, any character χ : A(O) → ℂ^{∗} is defined by the formula χ(p) = e^{−2πips/d}, where 0 ≤ s < d. We will denote this character by χ_{s}.

9.9 The action of F_{n}^{∗} on irreducible objects

Obviously, the center Z of G acts on F_{n}^{∗}(M) by z → z^{−sN/d}. Thus, a necessary condition for F_{n}^{∗}(M(O, χ_{s})) to be nonzero is

n = N (p + \frac{s}{d}) ,  (51)

where p is a nonnegative integer.

Our main result in this section is the following theorem.

Theorem 68. The functor F_{n}^{∗} maps irreducible objects into irreducible ones or zero. Specifically, if condition (51) holds, then we have

F_{n}^{∗}(M(O, χ_{s})) = L(π(nμ(O)/N)),

the irreducible representation of H_{n}(k) whose lowest weight is the representation of S_{n} corresponding to the partition nμ(O)/N.

Remark 69. Here if μ is a partition and c ∈ ℚ is a rational number, then we denote by cμ the partition whose parts are cμ_{i}, provided that these numbers are all integers. In our case, this integrality condition holds since all parts of μ(O) are divisible by d.

Corollary 70. Let λ be a partition of n into at most N parts. Let M = M(O_{μ}, χ_{s}), and assume that condition (51) is satisfied. Then

(M ⊗ V (λ))^{𝔤} = Hom_{S_{n}}(π(λ), L(π(nμ/N)))

as graded vector spaces.

This corollary allows us to express the characters of the irreducible Dmodules M(O, χ) in terms of characters of certain special lowest weight irreducible representations of H_{n}(k). We note that characters of lowest weight irreducible representations of rational Cherednik algebras of type A have been computed by Rouquier, [Rou05].

Remark 71. Note that Theorem 57 is the special case of Theorem 68 for O = {0}.

9.10 Proof of Theorem 68

Our proof of Theorem 68 is based on the following result of [GS05].

Theorem 72. Let k > 0. Then the functor V ↦ eV is an equivalence of categories between H_{n}(k)-modules and B_{n}(k)-modules.

Remark 73. We note that Theorem 72 is proved in [GS05] under the technical assumption k ∉ ℤ + 1/2. It was noticed by V. Ginzburg that this assumption is really unnecessary. Indeed, the only place where this assumption is used is in the proof of Lemma 3.5. Namely, it is used in the proof of this lemma that Hom between Verma modules over H_{n}(k) is isomorphic to Hom between the corresponding dual Specht modules, which is known, from [GGOR03], only for k ∉ ℤ + 1/2. However, it is sufficient for the proof of Lemma 3.5 of [GS05] to know just that the first Hom injects into the second one, which is known for all positive k thanks to a lemma by Opdam and Rouquier (Lemma 2.10 of [BEG03b]).

Theorem 72 implies the first statement of the theorem, i.e. that if (51) holds then F_{n}^{∗}(M(O_{μ}, χ_{s})) is irreducible. Indeed, it follows from Corollary 66 that eF_{n}^{∗}(M(O_{μ}, χ_{s})) is irreducible over B_{n}(k). Thus, it remains to find the lowest weight of F_{n}^{∗}(M(O_{μ}, χ_{s})).

Let μ = (μ_{1}, ..., μ_{N}) be a partition of N (μ_{i} ≥ 0). Let O_{μ} be the nilpotent orbit of 𝔤 corresponding to the partition μ. Denote by d the greatest common divisor of μ_{i}, and by m a divisor of d. Define the following function f on O_{μ} with values in ⊗_{i=1}^{N}S^{μ_{i}}ℂ^{N}:

f(X, ξ_{1}, ..., ξ_{N}) = ⋀_{i=1}^{N} ⋀_{j=0}^{μ_{i}−1} ξ_{i}X^{j},

ξ_{i} ∈ (ℂ^{N})^{∗} (here X^{j} ∈ M_{N}(ℂ) is the jth power of X, so ξ_{i}X^{j} ∈ ℂ^{N}).

Lemma 74. (i) For any X ∈ O_{μ}, f(X, …)^{1/m} is a polynomial in ξ_{1}, ..., ξ_{N}. Thus, f^{1/m} is a regular function on the universal cover Õ_{μ} of O_{μ} with values in ⊗_{i=1}^{N}S^{μ_{i}/m}ℂ^{N}.

(ii) For any X ∈ O_{μ}, the function f(X, …)^{1/m} generates a copy of the representation V (μ/m) inside ⊗_{i=1}^{N}S^{μ_{i}/m}ℂ^{N}.

(iii) Specifically, let the standard basis u_{1}, ..., u_{N} of (ℂ^{N})^{∗} be filled into the squares of the Young diagram of μ (filling the first column top to bottom, then the second one, etc.), and let X be the matrix J acting by the horizontal shift to the right on this basis. Then f(J, …)^{1/m} is a highest weight vector of the representation V (μ/m).

Proof. It is sufficient to prove (iii). Let μ^{∗} = (μ_{1}^{∗}, ..., μ_{N}^{∗}) be the conjugate partition. Let p_{j} be the number of times the part j occurs in this partition. Clearly, p_{j} is divisible by m. By looking at the matrix whose determinant is f, we see that we have, up to sign:

f(J, ξ_{1}, ..., ξ_{N}) = ∏_{j} Δ_{j}(ξ_{1}, ..., ξ_{N})^{p_{j}},

where Δ_{j} is the left upper j-by-j minor of the matrix (ξ_{1}, ..., ξ_{N}). Thus f^{1/m} = ∏_{j} Δ_{j}^{p_{j}/m} is clearly a highest weight vector of weight ∑_{j} p_{j}ϖ_{j}/m, where ϖ_{j} are the fundamental weights. But ∑ p_{j}ϖ_{j} = μ, so we are done. ⊓ ⊔

Corollary 75. The function f gives rise to a G-equivariant regular map f : Õ_{μ} → V (μ/d), whose image is the orbit of the highest weight vector. In particular, we have a G-equivariant inclusion of commutative algebras

f^{∗} : ⊕_{ℓ≥0}V (ℓμ/d)^{∗} → ℂ[Õ_{μ}].

Now let 0 ≤ s ≤ d − 1, and denote by ℂ[Õ_{μ}]_{s} the subspace of ℂ[Õ_{μ}], on which central elements z ∈ G act by z → z^{−s}. Then we have an inclusion

f^{∗} : ⊕_{ℓ:d^{−1}(ℓ−s)∈ℤ}V (ℓμ/d)^{∗} → ℂ[Õ_{μ}]_{s}.

Now recall that by construction, ℂ[Õ_{μ}]_{s} sits inside M = M(O_{μ}, χ_{s}) as a ℂ[O_{μ}]-submodule. In particular, the operators X_{i} act on the space (ℂ[Õ_{μ}]_{s} ⊗ (ℂ^{N})^{⊗n})^{𝔤}.

Let π(μ) be the representation of S_{n} corresponding to μ, and regard V (λ)⊗ π(λ), for any partition λ of n, as a subspace of (ℂ^{N})^{⊗n} using the Weyl duality. Then for any u ∈ π(nμ/N), we can define the element a(u) ∈ F_{n}^{∗}(M) by a(u) = f_{n}^{∗} ⊗ u, where f_{n}^{∗} ∈ ℂ[Õ_{μ}]_{s} ⊗ V (nμ/N) is the homogeneous part of f^{∗} of degree n.

Lemma 76. a(u) is annihilated by the elements y_{i} of H_{n}(k).

Proof. We need to show that the operators X_{i} (or, equivalently, the elements x_{i} ∈ H_{n}(k)) annihilate a(u) ∈ F_{n}(M). Since a(u) is G-invariant, it is sufficient to prove the statement at the point X = J. This boils down to showing that for any j not exceeding the number of parts of μ (i.e. j ≤ μ_{1}^{∗}), the application of J in any component annihilates the element Δ_{j}(ξ_{1}, ..., ξ_{N}) ∈ ∧^{j}ℂ^{N} ⊂ (ℂ^{N})^{⊗j}. This is clear, since the first μ_{1}^{∗} columns of J are zero. ⊓ ⊔

This implies that the lowest weight of F_{n}^{∗}(M(O_{μ}, χ_{s})) is π(nμ/N), as desired. The theorem is proved.

Remark 77. Here is another, short proof of Theorem 68 for n = N. We have

e_{−}F_{N}^{∗}(M(O, 1)) = F(M(O, 1))^{G}.

According to [Lev98, LS97],

F(M(O, 1))^{G} = (ℂ[𝔥] ⊗ π(μ(O)))^{S_{N}}

as a module over D(𝔥)^{W} = e_{−}H_{N}(1)e_{−}. Thus, e_{−}F_{N}^{∗}(M(O, 1)) = e_{−}L(π(μ(O))) as e_{−}H_{N}(1)e_{−}-modules. But the functor V → e_{−}V is an equivalence of categories H_{N}(1)-mod → e_{−}H_{N}(1)e_{−}-mod (see [BEG03b]). Thus, F_{N}^{∗}(M(O, 1)) = L(π(μ(O))) as H_{N}(1)-modules, as desired.

9.11 The support of L(π(nμ/N))

Corollary 78. Let μ be a partition of N such that nμ_{i}/N are integers. Then the support of the representation L(π(nμ/N)) of H_{n}(N/n) as a module over ℂ[x_{1}, ..., x_{n}] is contained in the union of S_{n}-translates of E_{q}, q = GCD(n, N).

Proof. This follows from Theorem 68 and Proposition 67. ⊓ ⊔

We note that in the case when μ = (N), Corollary 78 follows from Theorem 3.2 from [CE03].

9.12 The cuspidal case

An interesting special case of Theorem 68 is the cuspidal case. In this case N and n are relatively prime, d = N (i.e., O is the open orbit), and s is relatively prime to N.

Here is a short proof of Theorem 68 in the cuspidal case.

Since the Fourier transform of M(O, χ_{s}) in the cuspidal case is supported on the nilpotent cone, F_{n}^{∗}(M(O, χ_{s})) belongs not only to the category O generated by lowest weight modules, but also to the “dual” category O_{−} generated by highest weight modules over H_{n}(k). Thus, by the results of [BEG03a], F_{n}^{∗}(M(O, χ_{s})) is a multiple of the unique finite dimensional irreducible H_{n}(k)- module L(ℂ) = Y (N, n), of dimension N^{n−1}. But this multiple must be a single copy by Corollary 66, so the theorem is proved.

Theorem 68 implies the following formula for the characters of the cuspidal D-modules M(O, χ_{s}).

Let μ be a dominant integral weight for 𝔤, such that the center Z of G acts on V (μ) via z → z^{s} = z^{n}. Let ρ be the half-sum of positive roots of 𝔤. Let K_{s}(μ) = (M(O, χ_{s}) ⊗ V (μ))^{𝔤} be the isotypic components of M(O, χ_{s}).

Theorem 79. We have

Tr_{|K_{s}(μ)}(q^{2H}) = \frac{q − q^{−1}}{q^{N} − q^{−N}} φ_{μ}(q),

where

φ_{μ}(q) := ∏_{1≤p<r≤N} \frac{q^{μ_{r}−μ_{p}+r−p} − q^{μ_{p}−μ_{r}+p−r}}{q^{r−p} − q^{p−r}} = χ_{V (μ)}(q^{2ρ}),

where χ_{V (μ)} is the character of V (μ). In particular,

dim K_{s}(μ) = \frac{1}{N} ∏_{1≤p<r≤N} \frac{μ_{r} − μ_{p} + r − p}{r − p} = \frac{1}{N} dim V (μ).

Proof. We extend the representation V (μ) to GL_{N}(ℂ) by setting z → z^{n} for all scalar matrices z, so that its GL_{N}(ℂ)-highest weight is

μ̃ := (μ_{1} + n/N, ..., μ_{N} + n/N).

Note that we automatically have μ_{i} + n/N ∈ ℤ. Assume that n is so big that μ̃ is a partition of n (i.e., μ_{i} + n/N ≥ 0).

It follows from the results of [BEG03a] that the character of the irreducible representation L(ℂ) of the rational Cherednik algebra H_{n}(k), k = N/n, is given by the formula

Tr_{|L(ℂ)}(gq^{2h}) = \frac{q − q^{−1}}{q^{N} − q^{−N}} \frac{det(q^{−N} − q^{N}g)}{det(q^{−1} − qg)}, g ∈ S_{n},  (52)

where the determinants are taken in ℂ^{n}.

Let us equip ℂ^{N} with the structure of an irreducible representation of 𝔰𝔩_{2} with basis e, f, h. Let g ∈ S_{n}. Then

Tr_{|Hom_{Sn} (π(μ̃),(ℂ^{N})^{⊗n})}(q^{h}) = Tr_{|V (μ)}(q^{2ρ}) = φ_{μ}(q),

by the Weyl character formula. On the other hand, it is easy to show that

Tr_{|(ℂ^{N})^{⊗n}}(gq^{h}) = \frac{det(q^{−N} − q^{N}g)}{det(q^{−1} − qg)}.

Thus,

Tr_{|Hom_{Sn} (π(μ̃),L(ℂ))}(q^{2h}) = \frac{q − q^{−1}}{q^{N} − q^{−N}} Tr_{|Hom_{Sn} (π(μ̃),(ℂ^{N})^{⊗n})}(q^{h}) = \frac{q − q^{−1}}{q^{N} − q^{−N}} φ_{μ}(q).

By Theorem 68 and Weyl duality, this implies that

Tr_{|(M(O,χ_{s})⊗V (μ))^{𝔤}}(q^{2H}) = \frac{q − q^{−1}}{q^{N} − q^{−N}} φ_{μ}(q),

as desired. ⊓ ⊔ Example 80. Let N = 2, s = 1. In this case Theorem 79 gives us the following decomposition of M(O, χ_{s}):

M(O, χ_{s}) = ⊕_{j≥1}N_{j} ⊗ V_{2j−1},

where V_{j} is the irreducible representation of 𝔰𝔩_{2} of dimension j + 1, and the spaces N_{j} satisfy the equation

Tr_{|N_{j}}(q^{2H}) = \frac{q^{2j} − q^{−2j}}{q^{2} − q^{−2}} .

This shows that N_{j} = V_{j−1} as a representation of the 𝔰𝔩_{2}-subalgebra spanned by E, F, H, which commutes with 𝔤.

9.13 The case of general orbits

Let W = S_{N} the Weyl group of G, λ ∈ 𝔥/W, and N_{λ} be the closure in 𝔤 of the adjoint orbit of a regular element of 𝔤 whose semisimple part is λ. Denote by D_{G}(N_{λ}) the category of G-equivariant D-modules on G which are concentrated on N_{λ}. We also let O_{λ} be the category of finitely generated H_{n}(k)-modules in which the subalgebra ℂ[y_{1}, ..., y_{n}]^{S_{n}} acts through the character λ. Then one can show, similarly to the above, that the functor F_{n}^{∗} restricts to a functor F_{n,λ}^{∗} : D_{G}(N_{λ}) → O_{λ}. The functor considered above is F_{n,0}^{∗}. We plan to study the functor F_{n,λ}^{∗} for general λ in a future work.

9.14 The trigonometric case

Our results about rational Cherednik algebras can be extended to the trigonometric case. For this purpose, D-modules on the Lie algebra 𝔤 should be replaced with D-modules on the group G. Let us describe this generalization.

First, let us introduce some notation. As above, we let G = SL_{N}(ℂ). For b ∈ 𝔤, let L_{b} be the right invariant vector field on G equal to b at the identity element; that is, L_{b} generates the group of left translations by e^{tb}. As before, we let k = N/n.

Now let M be a D-module on G. Similarly to the above, we define F_{n}(M) to be the space

F_{n}(M) = (M ⊗ (ℂ^{N})^{⊗n})^{G},

where G acts on itself by conjugation.

Consider the operators X_{i}, Y_{i}, i = 1, ..., n, on F_{n}(M), defined by the formulas similar to the rational case:

X_{i} = ∑_{j,l} A_{jl} ⊗ (E_{lj})_{i}, Y_{i} = \frac{N}{n} ∑_{p} L_{b_{p}} ⊗ (b_{p})_{i},

where A_{jl} is the jl-th matrix element of A ∈ G regarded as the multiplication operator in M by a regular function on G.

Proposition 81. The operators X_{i}, Y_{i} satisfy the following relations:

∏_{i} X_{i} = 1, ∑_{i} Y_{i} + k ∑_{i<j} s_{ij} = 0,

s_{ij}X_{i} = X_{j}s_{ij}, s_{ij}Y_{i} = Y_{j}s_{ij}, [s_{ij}, X_{l}] = [s_{ij}, Y_{l}] = 0,

[X_{i}, X_{j}] = 0, [Y_{i}, Y_{j}] = ks_{ij}(Y_{i} − Y_{j}),

[Y_{i}, X_{j}] = (ks_{ij} − \frac{1}{n}) X_{j},

where i, j, l denote distinct indices.

Proof. Straightforward computation. ⊓ ⊔

Corollary 82. The operators Ȳ_{i} = Y_{i} + k ∑_{j<i} s_{ij} pairwise commute.

The relations of Proposition 81 are nothing but the defining relations of the degenerate double affine Hecke algebra of type A_{n−1}, which we will denote H_{n}^{tr}(k) (where “tr” stands for trigonometric, to illustrate the fact that this algebra is a trigonometric deformation of the rational Cherednik algebra H_{n}(k)). Thus we have defined an exact functor F_{n} : D(G) → H_{n}^{tr}(k)-mod. As before, it is sufficient to consider the restriction of this functor to the category of equivariant finitely generated D-modules, D_{G}(G).

This allows us to generalize much of our story for rational Cherednik algebras to the trigonometric case. In particular, let U be the unipotent variety on G, and D_{G}(U) be the category of finitely generated G-equivariant D-modules on G concentrated on U. If we restrict the functor F_{n} to this category, we get a situation identical to that in the rational case. Indeed, one can show that for any M in this category, F_{n}(M) belongs to the category O_{−}^{tr} of finitely generated modules over H_{n}^{tr}(k) which are locally unipotent with respect to the action of X_{i}. The latter category is equivalent to the category O_{−} over the rational Cherednik algebra H_{n}(k), because the completion of H_{n}^{tr}(k) with respect to the ideal generated by X_{i} − 1 is isomorphic to the completion of H_{n}(k) with respect to the ideal generated by x_{i}. On the other hand, the exponential map identifies the categories D_{G}(U) and D_{G}(N). It is clear that after we make these two identifications, the functor F_{n} becomes the functor F_{n} in the rational case that we considered above.

On the other hand, because of the absence of Fourier transform on the group (as opposed to Lie algebra), the trigonometric story is richer than the rational one. Namely, we can consider another subcategory of D_{G}(G), the category of character sheaves. By definition, a character sheaf on G is an object M in D_{G}(G) which is locally finite with respect to the action of the algebra of biinvariant differential operators, U(𝔤)^{G}. This category is denoted by Char(G). It is known that one has a decomposition

Char(G) = ⊕_{λ∈T^{∨}/W} Char_{λ}(G),

where T^{∨} is dual torus, and Char_{λ}(G) the category of those M ∈ D_{G}(G) for which the generalized eigenvalues of U(𝔤)^{G} (which we identify with U(𝔥)^{W} via the Harish-Chandra homomorphism) project to λ under the natural projection 𝔥^{∗} → T^{∨}.

On the other hand, one can define the category Rep_{Y −fin}(H_{n}^{tr}(k)) of modules over H_{n}^{tr}(k) on which the commuting elements Ȳ_{i} act in a locally finite manner. We have a similar decomposition

Rep_{Y −fin}(H_{n}^{tr}(k)) = ⊕_{λ∈T^{∨}/W} Rep_{Y −fin}(H_{n}^{tr}(k))_{λ},

where Rep_{Y −fin}(H_{n}^{tr}(k))_{λ} is the subcategory of all objects where the generalized eigenvalues of Ȳ_{i} project to λ ∈ T^{∨}/W. Then one can show, similarly to the rational case, that the functor F_{n} gives rise to the functors

F_{n,λ} : Char_{λ}(G) → Rep_{Y −fin}(H_{n}^{tr}(k))_{λ}

for each λ ∈ T^{∨}/W. The most interesting case is λ = 0 (unipotent character sheaves). We plan to study these functors in subsequent works.

9.15 Relation with the Arakawa-Suzuki functor

Note that the elements Y_{i} and s_{ij} generate the degenerate affine Hecke algebra H_{n} of Drinfeld and Lusztig (of type A_{n−1}). To define the action of this algebra on F_{n}(M) = (M ⊗ (ℂ^{N})^{⊗n})^{𝔤} by the formula of Proposition 81, we only need the action of the operators L_{b}, b ∈ 𝔤 in M. So M can be taken to be an arbitrary 𝔤-bimodule which is locally finite with respect to the diagonal action of 𝔤 (in this case, ∑_{i} Y_{i} + ∑_{i<j} s_{ij} is a central element which does not necessarily act by zero, so we get a representation of a central extension H̃_{n} of H_{n}). In particular, we have an exact functor F_{n} : HC(𝔤) → H̃_{n}-mod from the category of Harish-Chandra bimodules over 𝔤 to the category of finite dimensional representations of the degenerate affine Hecke algebra H̃_{n}. This functor was essentially considered in [AS98] (where it was applied to the Harish-Chandra modules of the form M = Hom_{𝔤−finite}(M_{1}, M_{2}), where M_{1} and M_{2} are modules from category O over 𝔤). We note that the paper [AST96] describes the extension of this construction to affine Lie algebras, which yields representations of degenerate double affine Hecke algebras.

9.16 Directions of further study

In conclusion we would like to discuss (in a fairly speculative manner) several directions of further study and generalizations (we note that these generalizations can be combined with each other).

1. The q-case: the group G is replaced with the corresponding quantum group, D-modules with q-D-modules, and degenerate double affine Hecke algebras with the usual double affine Hecke algebras (defined by Cherednik). It is especially interesting to consider this generalization if q is a root of unity.

2. The quiver case. This generalization was suggested by Ginzburg, and will be studied in his subsequent work with the third author. In this case, one has a finite subgroup Γ ⊂ SL_{2}(ℂ), and one should consider equivariant Dmodules on the representation space of the affine quiver attached to Γ (with some orientation). Then there should exist an analog of the functor F_{n}, which takes values in the category of representations of an appropriate symplectic reflection algebra for the wreath product S_{n}⋉Γ^{n}, [EG02] (or, equivalently, the Gan-Ginzburg algebra, [GG05]). This generalization should be especially nice in the case when Γ is a cyclic group, when the symplectic reflection algebra is a Cherednik algebra for a complex reflection group, and one has the notion of category O for it.

3. The symmetric space case. This is the trigonometric version of the previous generalization for Γ = ℤ/2. In this generalization one considers (monodromic) equivariant D-modules on the symmetric space GL_{p+q}(ℂ)/(GL_{p} × GL_{q})(ℂ) (see [Gin89]), and one expects a functor from this category to the category of representations of an appropriate degenerate double affine Hecke algebra of type C^{∨}C_{n}. This functor should be related, similarly to the previous subsection, to an analog of the Arakawa-Suzuki functor, which would attach to a Harish-Chandra module for the pair (GL_{p+q}(ℂ), GL_{p}(ℂ) × GL_{q}(ℂ)), a finite dimensional representation of the degenerate double affine Hecke algebra of type BC_{n}.

A

Let O be the ring ℂ[[u_{1}, ..., u_{n}]][ℓ_{1}, ..., ℓ_{n}]. Define commuting derivations D_{i} of O by D_{i}(u_{j}) = δ_{ij}u_{i}, D_{i}(ℓ_{j}) = δ_{ij} (we will later think of ℓ_{i} and D_{i} as logu_{i} and u_{i} \frac{∂}{∂u_{i}}).

We set O_{+} := 𝔪[ℓ_{1}, ..., ℓ_{n}], where 𝔪 = Ker(ℂ[[u_{1}, ..., u_{n}]] → ℂ) is the augmentation ideal. Let A = ⊕_{k≥0}A_{k} be a graded ring with finite dimensional homogeneous components.

Proposition 83. Let X_{i}(u_{1}, ..., ℓ_{n}) ∈ ˆ⊕_{k>0}(A_{k} ⊗ O_{+}) be such that D_{i}(X_{j}) = D_{j}(X_{i}). Then there exists a unique F (u_{1}, ..., ℓ_{n}) ∈ ˆ⊕_{k>0}(A_{k} ⊗ O_{+}) such that D_{i}(F) = X_{i} for i = 1, ..., n.

Let us say that f ∈ O has radius of convergence R > 0 if f = ∑_{k_{1},...,k_{n}≥0} f_{k_{1},...,k_{n}}(u_{1}, ..., u_{n})ℓ_{1}^{k_{1}}...ℓ_{n}^{k_{n}}, where each f_{k_{1},...,k_{n}}(u_{1}, ..., u_{n}) converges for |u_{1}|, ..., |u_{n}| ≤ R. Then if X_{1}, ..., X_{n} have radius of convergence R, so does F.

Proof. For each i, D_{i} restricts to an endomorphism of O_{+}; one checks that ∩_{i=1}^{n}Ker(D_{i} : O_{+} → O_{+}) = 0 which implies the uniqueness. To prove the existence, we work by induction. One proves that D_{n} : O_{+} → O_{+} is surjective, and its kernel is 𝔪_{n−1}[ℓ_{1}, ..., ℓ_{n−1}], where 𝔪_{n−1} = Ker(ℂ[[u_{1}, ..., u_{n−1}]] → ℂ). Let G be a solution of D_{n}(G) = X_{n}, then the system D_{i}(F′) = X_{i}−D_{i}(G) (i = 1, ..., n) is compatible, which implies D_{n}(X′_{i}) = 0, where X′_{i} := X_{i} −D_{i}(G), so X′_{i} ∈ ⊕_{k>0}(A_{k} ⊗ O_{+}^{(n−1)}), where O_{+}^{(n−1)} is the analogue of O_{+} at order n − 1. Hence the system D_{i}(F′) = X_{i} − D_{i}(G) (i = 1, ..., n − 1) is compatible and we may apply to it the result at order n − 1 to obtain a solution F′. Then a solution of D_{i}(F) = X_{i} is F′ + G.

Let D : uℂ[[u]] → uℂ[[u]] be the map u\frac{∂}{∂u} and let I := D^{−1}. The map D_{1} : uℂ[[u]][ℓ] → uℂ[[u]][ℓ] is bijective and its inverse is given by D_{1}^{−1}(F (u)ℓ^{a}) = ∑_{k=0}^{a}(−1)^{k}a(a − 1)...(a − k + 1)(I^{k+1}(F))(u)ℓ^{a−k}.

We have O_{+} = O^{(n−1)} ˆ⊗u_{n}ℂ[[u_{n}]][ℓ_{n}]⊕𝔪^{(n−1)} ˆ⊗ℂ[ℓ_{n}] (where O^{(n−1)}, 𝔪^{(n−1)} are the analogues of O, 𝔪 at order n − 1, ˆ⊗ is the completed tensor product). The endomorphism D_{n} preserves this decomposition and a section of D_{n} is given by (id ⊗ D_{1}^{−1}) ⊕ (id ⊗ J), where J ∈ End(ℂ[ℓ]) is a section of ∂/∂ℓ.

It follows from the fact that I preserves the radius of convergence of a series that the same holds for the section of D_{n} defined above. One then follows the above construction of a solution X of D_{i}(X) = X_{i} and uses the fact that D_{i} also preserves the radius of convergence to show by induction that X has radius R if the X_{i} do. ⊓ ⊔

Proposition 84. Let X_{i}(u_{1}, ..., ℓ_{n}) ∈ ˆ⊕_{k>0}(A_{k} ⊗ O_{+}) be such that D_{i}(X_{j}) − D_{j}(X_{i}) = [X_{i}, X_{j}]. Then there exists a unique F (u_{1}, ..., ℓ_{n}) ∈ 1 + ˆ⊕_{k>0}(A_{k} ⊗ O_{+}) such that D_{i}(F) = X_{i}F for i = 1, ..., n. If the X_{i} have radius R, then so does F.

Proof. Let us prove the uniqueness. If F, F′ are two solutions, then F^{−1}F′ is a constant (as ∩_{i=0}^{n}Ker(D_{i} : O → O) = 0), and it also belongs to 1 + ˆ⊕_{k>0}(A_{k} ⊗ O_{+}), which implies that F = F′. To prove the existence, one sets F = 1 + f_{1} + f_{2} + ..., X_{i} = x_{1}^{(i)} + ..., where f_{k}, x_{k}^{(i)} ∈ A_{k} ⊗ O_{+} and solves by induction the system D_{i}(f_{k}) = x_{1}^{(i)}f_{k−1} + ... + x_{k}^{(i)} using Proposition 83. ⊓ ⊔

Proposition 85. Let C_{i}(u_{1}, ..., u_{n}) ∈ ˆ⊕_{k>0}A_{k}[[u_{1}, ..., u_{n}]] (i = 1, ..., n) be such that u_{i}∂_{u_{i}}(C_{j}) − u_{j}∂_{u_{j}}(C_{i}) = [C_{i}, C_{j}] for any i, j. Assume that the series C_{i} have radius R. Then there exists a unique solution of the system u_{i}∂_{u_{i}}(X) = C_{i}X, analytic in the domain {u||u| ≤ R, u ∉ ℝ_{−}}^{n}, such that the ratio (u_{1}^{C_{0}^{1}}...u_{n}^{C_{0}^{n}})^{−1}X(u_{1}, ..., u_{n}) (we set C_{0}^{i} := C_{i}(0, ..., 0)) has the form 1 + ∑_{k>0} ∑_{a_{1},...,a_{n},i} r_{k}^{a_{1},...,a_{n},i}(u_{1}, ..., u_{n}) (the second sum is finite for any k), r_{k}^{a_{1},...,a_{n},i} has degree k, a_{i} ≥ 0, i ∈ {1, ..., n}, and r_{k}^{a_{1},...,a_{n},i}(u_{1}, ..., u_{n}) = O(u_{i}(logu_{1})^{a_{1}}...(logu_{n})^{a_{n}}).

The same is then true of the ratio X(u_{1}, ..., u_{n})(u_{1}^{C_{0}^{1}}...u_{n}^{C_{0}^{n}})^{−1}; we write X(u_{1}, ..., u_{n}) ≃ u_{1}^{C_{0}^{1}}...u_{n}^{C_{0}^{n}}.

Proof. Let us show the existence of X. The compatibility condition implies that [C_{0}^{i}, C_{0}^{j}] = 0. If we set Y (u_{1}, ..., u_{n}) := (u_{1}^{C_{0}^{1}}...u_{n}^{C_{0}^{n}})^{−1}X(u_{1}, ..., u_{n}), then X is a solution iff Y is a solution of u_{i}∂_{u_{i}}(Y) = exp(− ∑_{j=1}^{n}(logu_{j})C_{j}^{0})(C_{i} − C_{i}^{0}) · Y.

Let us set X_{i}(u_{1}, ..., ℓ_{n}) := exp(− ∑_{j=1}^{n} ℓ_{j}C_{j}^{0})(C_{i}(u_{1}, ..., u_{n}) − C_{i}(0, ..., 0)), then X_{i}(u_{1}, ..., ℓ_{n}) ∈ ˆ⊕_{k>0}(A_{k} ⊗ O_{+}). We then apply Proposition 84 and find a solution Y ∈ 1 + ˆ⊕_{k>0}A_{k} ⊗ O_{+} of D_{i}(Y) = X_{i}Y. Let Y_{k} be the component of Y of degree k. Since Y has radius R, the replacement ℓ_{i} = logu_{i} in Y_{k} for u_{i} ∈ {u||u| ≤ R, u ∉ ℝ_{−}} gives an analytic function on {u||u| ≤ R, u ∉ ℝ_{−}}^{n}. Moreover, O_{+} = ∑_{i=1}^{n} u_{i}ℂ[[u_{1}, ..., u_{n}]][ℓ_{1}, ..., ℓ_{n}], which gives a decomposition Y_{k} = ∑_{i,a_{1},...,a_{n}} u_{i}ℓ_{1}^{a_{1}}...ℓ_{n}^{a_{n}}y_{i,a_{1},...,a_{n}}^{k}(u_{1}, ..., u_{n}) and leads (after substitution ℓ_{i} = logu_{i}) to the above estimates.

The ratio X(u_{1}, ..., u_{n})(u_{1}^{C_{0}^{1}}...u_{n}^{C_{0}^{n}})^{−1} is then 1+exp(∑_{j} C_{0}^{j}logu_{j})(Y(u_{1}, ..., u_{n})− 1); the term of degree k has finitely many contributions to which we apply the above estimates.

Let us prove the uniqueness of X. Any other solution has the form X = X(1 + c_{k} + ...) where c_{j} ∈ A_{j}, and c_{k} ≠ 0. Then the degree k term is transformed by the addition of c_{k}, which cannot be split as a sum of terms in the various O(u_{i}(logu_{1})^{a_{1}}...(logu_{n})^{a_{n}}). ⊓ ⊔

Acknowledgments. This project started in June 2005, while the three authors were visiting ETH; they would like to thank Giovanni Felder for his kind invitations.

P.E. is deeply grateful to Victor Ginzburg for a lot of help with proofs of the main results in the part about equivariant D-modules, and for explanations on the Gan-Ginzburg functors. Without this help, this part could not have been written. P.E. is also very grateful to G. Lusztig for explaining to him the proof of Theorem 61, and to V. Ostrik and D. Vogan for useful discussions.

The work of D.C. has been partially supported by the European Union through the FP6 Marie Curie RTN ENIGMA (contract number MRTN-CT-2004-5652).

References

[AS98] Arakawa, T., Suzuki, T.: Duality between 𝔰l_{n}(ℂ) and the degenerate affine Hecke algebra. J. Algebra 209, no. 1, 288–304, (1998).

[AST96] Arakawa, T., Suzuki, T., Tsuchiya, A.: Degenerate double affine Hecke algebra and conformal field theory. In: Kashiwara, Masaki (ed.) et al., Topological field theory, primitive forms and related topics. Birkhäuser, Boston. Prog. Math. 160, 1–34, (1998).

[BK00] Bakalov, B., Kirillov (Jr.), A.: On the Lego-Teichmüller game. Transform. Groups 5, no. 3, 207–244, (2000).

[BEG03a] Berest, Y., Etingof, P., Ginzburg, V.: Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 19, 1053– 1088, (2003).

[BEG03b] Berest, Y., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118, no. 2, 279–337, [Ber98a] Bernard, D.: On the Wess-Zumino-Witten model on the torus. Nucl.

Phys. B 303, 77–93, (1998).

[Ber98b] Bernard, D.: On the Wess-Zumno-Witten model on Riemann surfaces. Nucl. Phys. B 309, 145–174, (1998).

[Bez94] Bezrukavnikov, R.: Koszul DG-algebras arising from configuration spaces. Geom. and Funct. Analysis 4, no. 2, 119–135, (1994).

[Bir69a] Birman, J.: On braid groups. Commun. Pure Appl. Math. 12, 41–79, (1969).

[Bir69b] Birman, J.: Mapping class groups and their relationship to braid groups. Commun. Pure Appl. Math. 12, 213–38 (1969).

[CE03] Chmutova, T., Etingof, P.: On some representations of the rational Cherednik algebra. Represent. Theory 7, 641–650, (2003).

[Che97] Cherednik, I.: Intertwining operators of double affine Hecke algebras. Selecta Math. (N.S.) 3, no. 4, 459–495, (1997).

[Che03] Cherednik, I.: Double affine Hecke algebras and difference Fourier transforms. Invent. Math. 152, no. 2, 213–303, (2003).

[Dri90a] Drinfeld, V.: On almost cocommutative Hopf algebras. Leningrad Math. J. 1, no. 2, 321–342, (1990).

[Dri90b] Drinfeld, V.: Quasi-Hopf algebras. Leningrad Math. J. 1, no. 6, 1419– 1457, (1990).

[Dri91] Drinfeld, V.: On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(ℚ̄/ℚ). Leningrad Math. J. 2, 829-860, (1991).

[EE05] Enriquez, B., Etingof, P.: Quantization of classical dynamical rmatrices with nonabelian base. Commun. Math. Phys. 254, no. 3, 603– 650, (2005).

[Eti94] Etingof, P.: Representations of affine Lie algebras, elliptic r-matrix systems, and special functions. Commun. Math. Phys. 159, no. 3, 471–502, (1994).

[EG02] Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, no. 2, 243–348, (2002).

[FW96] Felder, G., Wieczerkowski, C.: Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations. Commun. Math. Phys. 176, no. 1, 133–161, (1996).

[GG04] Gan, W.L., Ginzburg, V.: Almost-commuting variety, D-modules, and Cherednik Algebras. preprint math.RT/0409262.

[GG05] Gan, W.L., Ginzburg, V.: Deformed preprojective algebras and symplectic reflection algebras for wreath products. J. Algebra 283, no. 1, 350–363, (2005).

[Gin89] Ginzburg, V.: Admissible modules on a symmetric space. In: Orbites unipotentes et représentations. III. Orbites et faisceaux pervers. Astérisque 173-174, 199–255, (1989).

[GGOR03] Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On the category O for rational Cherednik algebras. Invent. Math. 154, no. 3, 617–651, (2003).

[GS05] Gordon, I., Stafford, J.T.: Rational Cherednik algebras and Hilbert schemes. Adv. Math. 198, no. 1, 222–274, (2005).

[HLS00] Hatcher, A., Lochak, P., Schneps, L.: On the Teichmüller tower of mapping class groups. J. Reine Angew. Math. 521, 1–24, (2000).

[Hes80] Hesselink, W.H.: Characters of the nullcone; Math. Ann. 252, no. 3, 179–182, (1980).

[HK84] Hotta, R., Kashiwara, M.: The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75, no. 2, 327–358, (1984).

[Jos97] Joseph, A.: On a Harish-Chandra homomorphism. C. R. Acad. Sci. Paris 324, 759–764, (1997).

[KZ84] Knizhnik V., Zamolodchikov, A.: Current algebras and the Wess-Zumino model in two dimensions. Nuclear Phys. B 247, 83–103, (1984).

[Koh83] Kohno, T.: On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. Nagoya Math. J. 92, 21–37, (1983).

[Kos63] Kostant, B.: Lie group representations on polynomial rings. Amer. J. Math. 85, 327–404, (1963).

[Kri94] Kriz, I.: On the rational homotopy type of configuration spaces. Annals Math. 139, 227–237, (1994).

[Lus81] Lusztig, G.: Singularities, character formulas, and a q-analog of weight multiplicities. In: Analysis and topology on singular spaces, II, III (Luminy, 1981), 208–229.

[Lus88] Lusztig, G.: Cuspidal local systems and graded Hecke algebras. I. Publ. Math. IHÉS 67, 145–202, (1988).

[Lus85] Lusztig, G.: Character sheaves. II. Adv. in Math. 57, no. 3, 226–265, (1985).

[Lev98] Levasseur, T.: Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra. Transformation Groups 3, 337–353, (1998).

[LS97] Levasseur, T., Stafford, J.T.: Semi-simplicity of invariant holonomic systems on a reductive Lie algebra. Amer. J. Math. 119, 1095–1117, (1997).

[Man05] Manin, Y.I.: Iterated integrals of modular forms and noncommutative modular symbols. In: Algebraic geometry and number theory. Birkhäuser, Boston. Progr. Math. 253, 565–597, (2006).

[Mir04] Mirkovic, I.: Character sheaves on reductive Lie slgebras. Mosc. Math. J. 4, no. 4, 897–910, (2004).

[Rou05] Rouquier, R.: q-Schur algebras and complex reflection groups. I. preprint

math.RT/0509252.

[Sul77] Sullivan, D.: Infinitesimal computations in topology. Publ. Math. IHÉS, 47, 260–331 (1977).