A generalization of the Capelli identity

E. Mukhin^{1}, V. Tarasov^{2}, and A. Varchenko^{3}

1 Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA,

mukhin@math.iupui.edu

2 Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA,

vtarasov@math.iupui.edu, and

St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia, vt@pdmi.ras.ru

3 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA, anv@email.unc.edu

Summary. We prove a generalization of the Capelli identity. As an application we obtain an isomorphism of the Bethe subalgebras actions under the (𝔤𝔩_{N}, 𝔤𝔩_{M}) duality.

To Yuri Manin on the occasion of 70-th birthday, with admiration.

1 Introduction

Let A be an associative algebra over complex numbers. Let A = (a_{ij})_{i,j=1}^{n} be an n × n matrix with entries in A. The row determinant of A is defined by the formula:

rdet(A) := ∑_{σ∈S_{n}} sgn(σ)a_{1σ_{1}} … a_{nσ_{n}}.

Let x_{ij}, i, j = 1, …, M, be commuting variables. Let ∂_{ij} = ∂/∂x_{ij},

E_{ij} = ∑_{a=1}^{M} x_{ia}∂_{ja}.  (1)

Let X = (x_{ij})_{i,j=1}^{M} and D = (∂_{ij})_{i,j=1}^{M} be M × M matrices.

The classical Capelli identity [C1] asserts the following equality of differential operators:

rdet (E_{ji} + (M − i)δ_{ij})_{i,j=1}^{M} = det(X) det(D).  (2)

This identity is a “quantization” of the identity

det(AB) = det(A) det(B)

for any matrices A, B with commuting entries.

The Capelli identity has the following meaning in the representation theory. Let ℂ[X] be the algebra of complex polynomials in variables x_{ij}. There are two natural actions of the Lie algebra 𝔤𝔩_{M} on ℂ[X]. The first action is given by operators from (1) and the second action is given by operators Ẽ_{ij} = ∑_{a=1}^{M} x_{ai}∂_{aj}. The two actions commute and the corresponding 𝔤𝔩_{M} ⊕ 𝔤𝔩_{M} action is multiplicity free.

It is not difficult to see that the right hand side of (2), considered as a differential operator on ℂ[X], commutes with both actions of 𝔤𝔩_{M} and therefore lies in the image of the center of the universal enveloping algebra U𝔤𝔩_{M} with respect to the first action. Then the left hand side of the Capelli identity expresses the corresponding central element in terms of U𝔤𝔩_{M} generators.

Many generalizations of the Capelli identity are known. One group of generalizations considers other elements of the center of U𝔤𝔩_{M}, called quantum immanants, and then expresses them in terms of 𝔤𝔩_{M} generators, see [C2], [N1],[O]. Another group of generalizations considers other pairs of Lie algebras in place of (𝔤𝔩_{M}, 𝔤𝔩_{M}), e.g. (𝔤𝔩_{M}, 𝔤𝔩_{N}), (𝔰𝔭_{2M}, 𝔤𝔩_{2}), (𝔰𝔭_{2M}, 𝔰𝔬_{N}), etc, see [MN], [HU]. The third group of generalizations produces identities corresponding not to pairs of Lie algebras, but to pairs of quantum groups [NUW] or superalgebras [N2].

In this paper we prove a generalization of the Capelli identity which seemingly does not fit the above classification.

Let z = (z_{1}, …, z_{N}), λ = (λ_{1}, …, λ_{M}) be sequences of complex numbers. Let Z = (z_{i}δ_{ij})_{ij=1}^{N}, Λ = (λ_{i}δ_{ij})_{ij=1}^{M} be the corresponding diagonal matrices. Let X and D be the M × N matrices with entries x_{ij} and ∂_{ij}, i = 1, …, M, j = 1, …, N, respectively. Let ℂ[X] be the algebra of complex polynomials in variables x_{ij}, i = 1, …, M, j = 1, …, N. Let E_{ij}^{(a)} = x_{ia}∂_{ja}, where i, j = 1, …, M, a = 1, …, N.

In this paper we prove that

∏_{a=1}^{N}(u − z_{a}) rdet ((∂_{u} − λ_{i})δ_{ij} − ∑_{a=1}^{N} \frac{E_{ji}^{(a)}}{u − z_{a}} )_{i,j=1}^{M} = rdet (\begin{matrix}u − Z & X^{t} \\ D & ∂_{u} − Λ \end{matrix}) . (3)

The left hand side of (3) is an M × M matrix while the right hand side is an (M + N) × (M + N) matrix.

Identity (3) is a “quantization” of the identity

det (\begin{matrix}A & B \\ C & D \end{matrix}) = det(A) det(D − CA^{−1}B)

which holds for any matrices A, B, C, D with commuting entries, for the case when A and D are diagonal matrices.

By setting all z_{i}, λ_{j} and u to zero, and N = M in (3), we obtain the classical Capelli identity (2), see Section 2.4.

Our proof of (3) is combinatorial and reduces to the case of 2×2 matrices. In particular, it gives a proof of the classical Capelli identity, which may be new.

We invented identity (3) to prove Theorem 6 below, and Theorem 6 in its turn was motivated by results of [MTV2]. In Theorem 6 we compare actions of two Bethe subalgebras.

Namely, consider ℂ[X] as a tensor product of evaluation modules over the current Lie algebras 𝔤𝔩_{M}[t] and 𝔤𝔩_{N}[t] with evaluation parameters z and λ, respectively. The action of the algebra 𝔤𝔩_{M}[t] on ℂ[X] is given by the formula

E_{ij} ⊗ t^{n} = ∑_{a=1}^{N} x_{ia}∂_{ja}z_{i}^{n},

and the action of the algebra 𝔤𝔩_{N}[t] on ℂ[X] is given by the formula

E_{ij} ⊗ t^{n} = ∑_{a=1}^{M} x_{ai}∂_{aj}λ_{i}^{n}.

In contrast to the previous situation, these two actions do not commute.

The algebra U𝔤𝔩_{M}[t] has a family of commutative subalgebras G(M, λ) depending on parameters λ and called the Bethe subalgebras. For a given λ, the Bethe subalgebra G(M, λ) is generated by the coefficients of the expansion of the expression

rdet ((∂_{u} − λ_{i})δ_{ij} − ∑_{a=1}^{N} ∑_{s=1}^{∞}(E_{ji}^{(a)} ⊗ t^{s})u^{−s−1})_{i,j=1}^{M}  (4)

with respect to powers of u and ∂_{u}, cf. Section 3. For different versions of definitions of Bethe subalgebras and relations between them, see [FFR], [T], [R], [MTV1].

Similarly, there is a family of Bethe subalgebras G(N, z) in U𝔤𝔩_{N}[t] depending on parameters z.

For fixed λ and z, consider the action of the Bethe subalgebras G(M, λ) and G(N, z) on ℂ[X] as defined above. In Theorem 6 we show that the actions of the Bethe subalgebras on ℂ[X] induce the same subalgebras of endomorphisms of ℂ[X].

The paper is organized as follows. In Section 2 we describe and prove formal Capelli-type identities and in Section 3 we discuss the relations of the identities to the Bethe subalgebras.

Acknowledgment. We thank the referee for bringing to our attention the papers [KS] and [GR] which relate the Capelli identity to Jordan algebras and “quasideterminants”, respectively.

Research of E.M. is supported in part by NSF grant DMS-0601005. Research of A.V. is supported in part by NSF grant DMS-0555327.

2 Identities

2.1 The main identity

We work over the field of complex numbers, however all results of this paper hold over any field of characteristic zero.

Let A be an associative algebra. Let A = (a_{ij})_{i,j=1}^{n} be an n × n matrix with entries in A. Define the row determinant of A by the formula:

rdet(A) := ∑_{σ∈S_{n}} sgn(σ)a_{1σ_{1}} … a_{nσ_{n}},

where S_{n} is the symmetric group on n elements.

Fix two natural numbers M and N and a complex number h ∈ ℂ. Consider noncommuting variables u, p_{u}, x_{ij}, p_{ij}, where i = 1, …, M, j = 1, …, N, such that the commutator of two variables equals zero except

[p_{u}, u] = h, [p_{ij}, x_{ij}] = h,

i = 1, …, M, j = 1, …, N.

Let X, P be two M × N matrices given by

X : = (x_{ij})_{i=1,...,M}^{j=1,...,N} , P : = (p_{ij})_{i=1,...,M}^{j=1,...,N} .

Let A_{h}^{(MN)} be the associative algebra whose elements are polynomials in p_{u}, x_{ij}, p_{ij}, i = 1, …, M, j = 1, …, N, with coefficients that are rational functions in u.

Let A^{(MN)} be the associative algebra of linear differential operators in u, x_{ij}, i = 1, …, M, j = 1, …, N, with coefficients in ℂ(u) ⊗ ℂ[X].

We often drop the dependence on M, N and write A_{h}, A for A_{h}^{(MN)} and A^{(MN)}, respectively.

For h ≠ 0, we have the isomorphism of algebras

ι_{h} : A_{h} → A ,  (5)
u, x_{ij} ↦ u, x_{ij} ,
p_{u}, p_{ij} ↦ h\frac{∂}{∂u}, h\frac{∂}{∂x_{ij}} .

Fix two sequences of complex numbers z = (z_{1}, …, z_{N}) and λ = (λ_{1}, …, λ_{M}).

Define the M ×M matrix G_{h} = G_{h}(M, N, u, p_{u}, z, λ, X, P) by the formula

G_{h} : = ( (p_{u} − λ_{i}) δ_{ij} − ∑_{a=1}^{N} \frac{x_{ja}p_{ia}}{u − z_{a}} )_{i,j=1}^{M} .  (6)

Theorem 1. We have

∏_{a=1}^{N}(u−z_{a}) rdet(G_{h}) = ∑_{A,B,|A|=|B|}(−1)^{|A|} ∏_{a∉B}(u−z_{a}) ∏_{b∉A}(p_{u}−λ_{b}) det(x_{ab})_{a∈A}^{b∈B} det(p_{ab})_{a∈A}^{b∈B},

where the sum is over all pairs of subsets A ⊂ {1, …, M}, B ⊂ {1, …, N} such that A and B have the same cardinality, |A| = |B|. Here the sets A, B inherit the natural ordering from the sets {1, …, M}, {1, …, N}. This ordering determines the determinants in the formula.

Theorem 1 is proved in Section 2.5.

2.2 A presentation as a row determinant of size M + N

Theorem 1 implies that the row determinant of G can be written as the row determinant of a matrix of size M + N.

Namely, let Z be the diagonal N × N matrix with diagonal entries z_{1}, …, z_{N}. Let Λ be the diagonal M × M matrix with diagonal entries λ_{1}, …, λ_{M}:

Z : = ( z_{i}δ_{ij} )_{i,j=1}^{N} , Λ : = ( λ_{i}δ_{ij} )_{i,j=1}^{M} .

Corollary 2. We have

∏_{a=1}^{N}(u − z_{a}) rdet G = rdet (\begin{matrix}u − Z & X^{t} \\ P & p_{u} − Λ \end{matrix}) ,

where X^{t} denotes the transpose of the matrix X.

Proof. Denote

W : = (\begin{matrix}u − Z & X^{t} \\ P & p_{u} − Λ \end{matrix}) ,

The entries of the first N rows of W commute. The entries of the last M rows of W also commute. Write the Laplace decomposition of rdet(W) with respect to the first N rows. Each term in this decomposition corresponds to a choice of N columns in the N ×(N +M) matrix (u−Z, X^{T}). We label such a choice by a pair of subsets A ⊂ {1, …, M} and B ⊂ {1, …, N} of the same cardinality. Namely, the elements of A correspond to the chosen columns in X^{T} and the elements of the complement to B correspond to the chosen columns in u − Z. Then the term in the Laplace decomposition corresponding to A and B is exactly the term labeled by A and B in the right hand side of the formula in Theorem 1. Therefore, the corollary follows from Theorem 1. □

Let A, B, C, D be any matrices with commuting entries of sizes N × N, N × M, M × N and M × M, respectively. Let A be invertible. Then we have the equality of matrices of sizes (M + N) × (M + N):

(\begin{matrix}A & B \\ C & D \end{matrix}) = (\begin{matrix}A & 0 \\ C & D − CA^{−1}B \end{matrix}) (\begin{matrix}1 & A^{−1}B \\ 0 & 1 \end{matrix})

and therefore

det (\begin{matrix}A & B \\ C & D \end{matrix}) = det(A) det(D − CA^{−1}B) .  (7)

The identity of Corollary 2 for h = 0 turns into identity (7) with diagonal matrices A and D. Therefore, the identity of Corollary 2 may be thought of as a “quantization” of identity (7) with diagonal A and D.

2.3 A relation between determinants of sizes M and N

Introduce new variables v, p_{v} such that [p_{v}, v] = h.

Let Ā_{h} be the associative algebra whose elements are polynomials in p_{u}, p_{v}, x_{ij}, p_{ij}, i = 1, …, M, j = 1, …, N, with coefficients in ℂ(u) ⊗ ℂ(v).

Let e : Ā_{h} → Ā_{h} be the unique linear map which is the identity map on the subalgebra of Ā_{h} generated by all monomials which do not contain p_{u} and p_{v} and which satisfy

e(ap_{u}) = e(a)v, e(ap_{v}) = e(a)u,

for any a ∈ Ā_{h}.

Let Ā be the associative algebra of linear differential operators in u, v, x_{ij}, i = 1, …, M, j = 1, …, N, with coefficients in ℂ(u) ⊗ ℂ(v) ⊗ ℂ[x_{ij}]. Then for h ≠ 0, we have the isomorphism of algebras extending the isomorphism (5):

ῑ_{h} : Ā_{h} → Ā,
u, v, x_{ij} ↦ u, v, x_{ij},
p_{u}, p_{v}, p_{ij} ↦ h\frac{∂}{∂u}, h\frac{∂}{∂v} , h\frac{∂}{∂x_{ij}} .

For a ∈ Ā and a function f(u, v) let a·f(u, v) denotes the function obtained by the action of a considered as a differential operator in u and v on the function f(u, v).

We have

ῑ_{h}(e(a)) = exp(−uv/h)ῑ_{h}(a) · exp(uv/h)

for any a ∈ Ā_{h} such that a does not depend on either p_{u} or p_{v}.

Define the N × N matrix H_{h} = H_{h}(M, N, v, p_{v}, z, λ, X, P) by

H_{h} := ((p_{v} − z_{i})δ_{ij} − ∑_{b=1}^{M} \frac{x_{bj}p_{bi}}{v − λ_{b}} )_{i,j=1}^{N},  (8)

cf. formula (6).

Corollary 3. We have

e( ∏_{a=1}^{N}(u − z_{a}) rdet(G_{h})) = e( ∏_{b=1}^{M}(v − λ_{b}) rdet(H_{h})).

Proof. Write the dependence on parameters of the matrix G: G_{h} = G_{h}(M, N, u, p_{u}, z, λ, X, P). Then

H_{h} = G_{h}(N, M, v, p_{v}, λ, z, X^{T} , P^{T}).

The corollary now follows from Theorem 1. □

2.4 A relation to the Capelli identity

In this section we show how to deduce the Capelli identity from Theorem 1.

Let s be a complex number. Let α_{s} : A_{h} → A_{h} be the unique linear map which is the identity map on the subalgebra of A_{h} generated by all monomials which do not contain p_{u}, and which satisfies

α_{s}(aup_{u}) = sα_{s}(a)

for any a ∈ Ā_{h}.

We have

ῑ_{h}(α_{s}(a)) = u^{−s/h}ῑ_{h}(a) · u^{s/h}

for any a ∈ Ā_{h}.

Consider the case z_{1} = ⋯ = z_{N} = 0 and λ_{1} = ⋯ = λ_{M} = 0 in Theorem 1.

Then it is easy to see that the row determinant rdet(G) can be rewritten in the following form

u^{M} rdet(G_{h}) = rdet (h(up_{u} − M + i)δ_{ij} − ∑_{a=1}^{N} x_{ja}p_{ia})_{i,j=1}^{M}.

Applying the map α_{s}, we get

α_{s}(u^{M} rdet(G_{h})) = rdet (h(s − M + i)δ_{ij} − ∑_{a=1}^{N} x_{ja}p_{ia})_{i,j=1}^{M}.

Therefore applying Theorem 1 we obtain the identity

rdet (h(s−M+i)δ_{ij}−∑_{a=1}^{N} x_{ja}p_{ia})_{i,j=1}^{M} = ∑_{A,B,|A|=|B|} (−1)^{|A|} ∏_{b=0}^{M−|A|−1} (s−bh) det(x_{ab})_{a∈A}^{b∈B} det(p_{ab})_{a∈A}^{b∈B}.

In particular, if M = N, and s = 0, we obtain the famous Capelli identity:

rdet ( ∑_{a=1}^{M} x_{ja}p_{ia} + h(M − i)δ_{ij})_{i,j=1}^{M} = det X det P.

If h = 0 then all entries of X and P commute and the Capelli identity reads det(XP) = det(X) det(P). Therefore, the Capelli identity can be thought of as a “quantization” of the identity det(AB) = det(A) det(B) for square matrices A, B with commuting entries.

2.5 Proof of Theorem 1

We denote

E_{ij,a} := x_{ja}p_{ia}/(u − z_{a}).

We obviously have

[E_{ij,a}, E_{kl,b}] = δ_{ab}(δ_{kj}(E_{il,a})′ − δ_{il}(E_{kj,a})′),

where the prime denotes the formal differentiation with respect to u.

Denote also F_{jk,a}^{1} = −E_{jk,a} and F_{jj,0}^{0} = (p_{u} − λ_{j}).

Expand rdet(G). We get an alternating sum of terms,

rdet(G_{h}) = ∑_{σ,a,c}(−1)^{sgn(σ)}F_{1σ(1),a(1)}^{c(1)}F_{2σ(2),a(2)}^{c(2)} … F_{Mσ(M),a(M)}^{c(M)},  (9)

where the summation is over all triples σ, a, c such that σ is a permutation of {1, …, M} and a, c are maps a : {1, …, M} → {0, 1, …, N}, c : {1, …, M} → {0, 1} satisfying: c(i) = 1 if σ(i) ≠ i; a(i) = 0 if and only if c(i) = 0.

Let m be a product whose factors are of the form f(u), p_{u}, p_{ij}, x_{ij} where f(u) are some rational functions in u. Then the product m will be called normally ordered if all factors of the form p_{u}, p_{ij} are on the right from all factors of the form f(u), x_{ij}. For example, (u − 1)^{−2}x_{11}p_{u}p_{11} is normally ordered and p_{u}(u − 1)^{−2}x_{11}p_{11} is not.

Given a product m as above, define a new normally ordered product : m : as the product of all factors of m in which all factors of the form p_{u}, p_{ij} are placed on the right from all factors of the form f(u), x_{ij}. For example, : p_{u}(u − 1)^{−2}x_{11}p_{11} := (u − 1)^{−2}x_{11}p_{u}p_{11}.

If all variables p_{u}, p_{ij} are moved to the right in the expansion of rdet(G) then we get terms obtained by normal ordering from the terms in (9) plus new terms created by the non-trivial commutators. We show that in fact all new terms cancel in pairs.

Lemma 4. For i = 1, …, M, we have

rdet(G_{h}) = ∑_{σ,a,c}(−1)^{sgn(σ)}F_{1σ(1),a(1)}^{c(1)} … F_{(i−1)σ(i−1),a(i−1)}^{c(i−1)}( : F_{iσ(i),a(i)}^{c(i)} … F_{Mσ(M),a(M)}^{c(M)} : ), (10)

where the sum is over the same triples σ, a, c as in (9).

Proof. We prove the lemma by induction on i. For i = M the lemma is a tautology. Assume it is proved for i = M, M − 1, …, j, let us prove it for i = j − 1.

We have

F_{(j−1)r,a}^{1} : F_{jσ(j),a(j)}^{c(j)} … F_{M,σ(M),a(M)}^{c(M)} :=  (11)

: F_{(j−1)r,a}^{1}F_{jσ(j),a(j)}^{c(j)} … F_{Mσ(M),a(M)}^{c(M)} : + ∑_{k} : F_{jσ(j),a(j)}^{c(j)} … (−E_{kr,a})′ … F_{Mσ(M),a(M)}^{c(M)} : ,

where the sum is over k ∈ {j, …, M} such that a(k) = a, σ(k) = j − 1 and c(k) = 1.

We also have

F_{(j−1)(j−1),0}^{0} : F_{jσ(j),a(j)}^{c(j)} … F_{Mσ(M),a(M)}^{c(M)} :=  (12)

: F_{(j−1)(j−1),0}^{0}F_{jσ(j),a(j)}^{c(j)} … F_{Mσ(M),a(M)}^{c(M)} : + ∑_{k} : F_{jσ(j),a(j)}^{c(j)} … (−E_{kσ(k),a(k)})′ … F_{Mσ(M),a(M)}^{c(M)} : ,

where the sum is over k ∈ {j, …, M} such that c(k) = 1.

Using (11), (12), rewrite each term in (10) with i = j. Then the k-th term obtained by using (11) applied to the term labeled by σ, c, a with c(j − 1) = 0 cancels with the k-th obtained by using (12) applied to the term labeled by σ̃, c̃, ã defined by the following rules.

σ̃(i) = σ(i) (i ≠ j − 1, k),  σ̃(j − 1) = j − 1,  σ̃(k) = σ(j − 1),
c̃(i) = c(i) (i ≠ j − 1),  c̃(j − 1) = 0,
ã(i) = a(i) (i ≠ j − 1),  ã(j − 1) = 0.

After this cancellation we obtain the statement of the lemma for i = j −1. □

Remark 5. The proof of Lemma 4 implies that if the matrix σG_{h} is obtained from G_{h} by permuting the rows of G_{h} by a permutation σ then rdet(σG_{h}) = (−1)^{sgn(σ)} rdet(G_{h}).

Consider the linear isomorphism ϕ_{h} : A_{h} → A_{0} which sends any normally ordered monomial m in A_{h} to the same monomial m in A_{0}.

By (10) with i = 1, the image ϕ_{h}(rdet(G_{h})) does not depend on h and therefore can be computed at h = 0. Therefore Theorem 1 for all h follows from Theorem 1 for h = 0. Theorem 1 for h = 0 follows from formula (7).

3 The (𝔤𝔩_{M}, 𝔤𝔩_{N}) duality and the Bethe subalgebras

3.1 Bethe subalgebra

Let E_{ij}, i, j = 1, …, M, be the standard generators of 𝔤𝔩_{M}. Let 𝔥 be the Cartan subalgebra of 𝔤𝔩_{M},

𝔥 = ⊕_{i=1}^{M}ℂ · E_{ii}.

We denote U𝔤𝔩_{M} the universal enveloping algebra of 𝔤𝔩_{M}.

For μ ∈ 𝔥^{∗}, and a 𝔤𝔩_{M} module L denote by L[μ] the vector subspace of L of vectors of weight μ,

L[μ] = {v ∈ L | hv = ⟨μ, h⟩ v for any h ∈ 𝔥}.

We always assume that L = ⊕_{μ}L[μ].

For any integral dominant weight Λ ∈ 𝔥^{∗}, denote by L_{Λ} the finitedimensional irreducible 𝔤𝔩_{M}-module with highest weight Λ.

Recall that we fixed sequences of complex numbers z = (z_{1}, …, z_{N}) and λ = (λ_{1}, …, λ_{M}). From now on we will assume that z_{i} ≠ z_{j} and λ_{i} ≠ λ_{j} if i ≠ j.

For i, j = 1, …, M,, a = 1, …, N, let E_{ji}^{(a)} = 1^{⊗(a−1)} ⊗ E_{ji} ⊗ 1^{⊗(N−a)} ∈ (U𝔤𝔩_{M})^{⊗N}.

Define the M × M matrix G̃ = G̃(M, N, z, λ, u) by

G̃(M, N, z, λ, u) := ((\frac{∂}{∂u} − λ_{i})δ_{ij} − ∑_{a=1}^{N} \frac{E_{ji}^{(a)}}{u − z_{a}} )_{i,j=1}^{M}.

The entries of G̃ are differential operators in u whose coefficients are rational functions in u with values in (U𝔤𝔩_{M})^{⊗N}.

Write

rdet(G̃(M, N, z, λ, u)) = \frac{∂^{M}}{∂u^{M}} +G̃_{1}(M, N, z, λ, u)\frac{∂^{M−1}}{∂u^{M−1}} +⋯+G̃_{M}(M, N, z, λ, u).

The coefficients G̃_{i}(M, N, z, λ, u), i = 1, …, M, are called the transfer matrices of the Gaudin model. The transfer matrices are rational functions in u with values in (U𝔤𝔩_{M})^{⊗N}.

The transfer matrices commute:

[G̃_{i}(M, N, z, λ, u), G̃_{j}(M, N, z, λ, v)] = 0,

for all i, j, u, v, see [T] and Proposition 7.2 in [MTV1].

The transfer matrices clearly commute with the diagonal action of 𝔥 on (U𝔤𝔩_{M})^{⊗N}.

For i = 1, …, M, it is clear that G̃_{i}(M, N, z, λ, u) ∏_{a=1}^{N}(u−z_{a})^{i} is a polynomial in u whose coefficients are pairwise commuting elements of (U𝔤𝔩_{M})^{⊗N}. Let G(M, N, z, λ) ⊂ (U𝔤𝔩_{M})^{⊗N} be the commutative subalgebra generated by the coefficients of polynomials G̃_{i}(M, N, z, λ, u) ∏_{a=1}^{N}(u − z_{a})^{i}, i = 1, …, M. We call the subalgebra G(M, N, z, λ) the Bethe subalgebra.

Let G(M, λ) ⊂ U𝔤𝔩_{M}[t] be the subalgebra considered in the introduction. Let U𝔤𝔩_{M}[t] → (U𝔤𝔩_{M})^{⊗N} be the algebra homomorphism defined by E_{ij} ⊗ t^{n} ↦ ∑_{a=1}^{N} E_{ij}^{(a)}z_{a}^{n}. Then the subalgebra G(M, N, z, λ) is the image of the subalgebra G(M, λ) under that homomorphism.

The Bethe subalgebra clearly acts on any N-fold tensor products of 𝔤𝔩_{M} representations.

Define the Gaudin Hamiltonians, H_{a}(M, N, z, λ) ⊂ (U𝔤𝔩_{M})^{⊗N}, a = 1, …, N, by the formula

H_{a}(M, N, z, λ) = ∑_{b=1,b≠a}^{N} \frac{Ω^{(ab)}}{z_{a} − z_{b}} + ∑_{b=1}^{M} λ_{b}E_{bb}^{(a)},

where Ω^{(ab)} := ∑_{i,j=1}^{M} E_{ij}^{(a)}E_{ji}^{(b)}.

Define the dynamical Hamiltonians H_{a}^{∨}(M, N, z, λ) ⊂ (U𝔤𝔩_{M})^{⊗N}, a = 1, …, M, by the formula

H_{a}^{∨}(M, N, z, λ) = ∑_{b=1, b≠a}^{M} \frac{(∑_{i=1}^{N} E_{ab}^{(i)})(∑_{i=1}^{N} E_{ba}^{(i)}) − ∑_{i=1}^{N} E_{aa}^{(i)}}{λ_{a} − λ_{b}} + ∑_{b=1}^{N} z_{b} E_{aa}^{(b)} .

It is known that the Gaudin Hamiltonians and the dynamical Hamiltonians are in the Bethe subalgebra, see e.g. Appendix B in [MTV1]:

H_{a}(M, N, z, λ) ∈ G(M, N, z, λ), H_{b}^{∨}(M, N, z, λ) ∈ G(M, N, z, λ),

a = 1, …, N, b = 1, …, M.

3.2 The (𝔤𝔩_{M}, 𝔤𝔩_{N}) duality

Let L_{•}^{(M)} = ℂ[x_{1}, …, x_{M}] be the space of polynomials of M variables. We define the 𝔤𝔩_{M}-action on L_{•}^{(M)} by the formula

E_{ij} ↦ x_{i} \frac{∂}{∂x_{j}} .

Then we have an isomorphism of 𝔤𝔩_{M} modules

L_{•}^{(M)} = ⨁_{m=0}^{∞} L_{m}^{(M)}

the submodule L_{m}^{(M)} being spanned by homogeneous polynomials of degree m. The submodule L_{m}^{(M)} is the irreducible 𝔤𝔩_{M} module with highest weight (m, 0, …, 0) and highest weight vector x_{1}^{m}.

Let L_{•}^{(M,N)} = ℂ[x_{11}, …, x_{1N}, …, x_{M1}, …, x_{MN}] be the space of polynomials of MN commuting variables.

Let π^{(M)} : (U𝔤𝔩_{M})^{⊗N} → End(L_{•}^{(M,N)}) be the algebra homomorphism defined by

E_{ij}^{(a)} ↦ x_{ia} \frac{∂}{∂x_{ja}} .

In particular, we define the 𝔤𝔩_{M} action on L_{•}^{(M,N)} by the formula

E_{ij} ↦ ∑_{a=1}^{N} x_{ia} \frac{∂}{∂x_{ja}} .

Let π^{(N)} : (U𝔤𝔩_{N})^{⊗M} → End(L_{•}^{(M,N)}) be the algebra homomorphism defined by

E_{ij}^{(a)} ↦ x_{ai} \frac{∂}{∂x_{aj}} .

In particular, we define the 𝔤𝔩_{N} action on L_{•}^{(M,N)} by the formula

E_{ij} ↦ ∑_{a=1}^{M} x_{ai} \frac{∂}{∂x_{aj}} .

We have isomorphisms of algebras,

(ℂ[x_{1}, …, x_{M}])^{⊗N}→ L_{•}^{(M,N)}, 1^{⊗(j−1)} ⊗ x_{i} ⊗ 1^{⊗(N−j)} ↦ x_{ij},
(ℂ[x_{1}, …, x_{N}])^{⊗M}→ L_{•}^{(M,N)}, 1^{⊗(i−1)} ⊗ x_{j} ⊗ 1^{⊗(M−i)} ↦ x_{ij}. (13)

Under these isomorphisms the space L_{•}^{(M,N)} is isomorphic to (L_{•}^{(M)})^{⊗N} as a 𝔤𝔩_{M} module and to (L_{•}^{(N)})^{⊗M} as a 𝔤𝔩_{N} module.

Fix n = (n_{1}, …, n_{N}) ∈ ℤ_{≥0}^{N} and m = (m_{1}, …, m_{M}) ∈ ℤ_{≥0}^{M} with ∑_{i=1}^{N} n_{i} = ∑_{a=1}^{M} m_{a}. The sequences n and m naturally correspond to integral 𝔤𝔩_{N} and 𝔤𝔩_{M} weights, respectively.

Let L_{m} and L_{n} be 𝔤𝔩_{N} and 𝔤𝔩_{M} modules, respectively, defined by the formulas

L_{m} = ⊗_{a=1}^{M}L_{m_{a}}^{(N)}, L_{n} = ⊗_{b=1}^{N}L_{n_{b}}^{(M)}.

The isomorphisms (13) induce an isomorphism of the weight subspaces,

L_{n}[m] ≃ L_{m}[n].  (14)

Under the isomorphism (14) the Gaudin and dynamical Hamiltonians interchange,

\begin{aligned}π^{(M)}H_{a}(M, N, z, λ) & = & π^{(N)}H_{a}^{∨}(N, M, λ, z) , \\ π^{(M)}H_{b}^{∨}(M, N, z, λ) & = & π^{(N)}H_{b}(N, M, λ, z) , \end{aligned}

for a = 1, …, N, b = 1, …, M, see [TV].

We prove a stronger statement that the images of 𝔤𝔩_{M} and 𝔤𝔩_{N} Bethe subalgebras in End(L_{•}^{(M,N)}) are the same.

Theorem 6. We have

π^{(M)}(G(M, N, z, λ)) = π^{(N)}(G(N, M, λ, z)).

Moreover, we have

∏_{a=1}^{N}(u − z_{a})π^{(M)} rdet(G̃(M, N, z, λ, u)) = ∑_{a=1}^{N} ∑_{b=1}^{M} A_{ab}^{(M)} u^{a} \frac{∂^{b}}{∂u^{b}} ,

∏_{b=1}^{M}(v − λ_{b})π^{(N)} rdet(G̃(N, M, λ, z, v)) = ∑_{a=1}^{N} ∑_{b=1}^{M} A_{ab}^{(N)} v^{b} \frac{∂^{a}}{∂v^{a}} ,

where A_{ab}^{(M)}, A_{ab}^{(N)} are linear operators independent on u, v, ∂/∂u, ∂/∂v and

A_{ab}^{(M)} = A_{ab}^{(N)}.

Proof. We obviously have

π^{(M)}(G̃(M, N, z, λ, u)) = ī_{h=1}(G_{h=1}),
π^{(N)}(G̃(N, M, λ, z, v)) = ī_{h=1}(H_{h=1}),

where G_{h=1} and H_{h=1} are matrices defined in (6) and (8). Then the coefficients of the differential operators ∏_{a=1}^{N}(u − z_{a})π^{(M)} rdet(G̃(M, N, z, λ, u)) and ∏_{b=1}^{M}(v − λ_{b})π^{(N)} rdet(G̃(N, M, λ, z, v)) are polynomials in u and v of degrees N and M, respectively, by Theorem 1. The rest of the theorem follows directly from Corollary 3. □

3.3 Scalar differential operators

Let w ∈ L_{n}[m] be a common eigenvector of the Bethe subalgebra G(M, N, z, λ). Then the operator rdet(G̃(M, N, z, λ, u)) acting on w defines a monic scalar differential operator of order M with rational coefficients in variable u. Namely, let D_{w}(M, N, λ, z) be the differential operator given by

D_{w}(M, N, z, λ, u) = \frac{∂^{M}}{∂u^{M}} +G̃_{1}^{w}(M, N, z, λ, u)\frac{∂^{M−1}}{∂u^{M−1}} +⋯+G̃_{M}^{w}(M, N, z, λ, u),

where G̃_{i}^{w}(M, N, z, λ, u) is the eigenvalue of the ith transfer matrix acting on the vector w:

G̃_{i}(M, N, z, λ, u)w = G̃_{i}^{w}(M, N, z, λ, u)w.

Using isomorphism (14), consider w as a vector in L_{m}[n]. Then by Theorem 6, w is also a common eigenvector for algebra G(N, M, λ, z). Thus, similarly, the operator rdet(G̃(N, M, λ, z, v)) acting on w defines a monic scalar differential operator of order N, D_{w}(N, M, λ, z, v).

Corollary 7. We have

∏_{a=1}^{N}(u − z_{a})D_{w}(M, N, z, λ, u) = ∑_{a=1}^{N} ∑_{b=1}^{M} A_{ab,w}^{(M)} u^{a} \frac{∂^{b}}{∂u^{b}} ,

∏_{b=1}^{M}(v − λ_{b})D_{w}(N, M, λ, z, v) = ∑_{a=1}^{N} ∑_{b=1}^{M} A_{ab,w}^{(N)} v^{b} \frac{∂^{a}}{∂v^{a}} ,

where A_{ab,w}^{(M)}, A_{ab,w}^{(N)} are numbers independent on u, v, ∂/∂u, ∂/∂v. Moreover,

A_{ab,w}^{(M)} = A_{ab,w}^{(N)}.

Proof. The corollary follows directly from Theorem 6. □

Corollary 7 was essentially conjectured in Conjecture 5.1 in [MTV2].

Remark 8. The operators D_{w}(M, N, z, λ) are useful objects, see [MV1], [MTV2], [MTV3]. They have the following three properties.

(i) The kernel of D_{w}(M, N, z, λ) is spanned by the functions p_{i}^{w}(u)e^{λ_{i}u}, i = 1, …, M, where p_{i}^{w}(u) is a polynomial in u of degree m_{i}.

(ii) All finite singular points of D_{w}(M, N, z, λ) are z_{1}, …, z_{N}. (iii) Each singular point z_{i} is regular and the exponents of D_{w}(M, N, z, λ) at

z_{i} are 0, n_{i} + 1, n_{i} + 2, …, n_{i} + M − 1.

A converse statement is also true. Namely, if a linear differential operator of order M has properties (i-iii), then the operator has the form D_{w}(M, N, z, λ) for a suitable eigenvector w of the Bethe subalgebra. This statement may be deduced from Proposition 9 below.

We discuss the properties of such differential operators in [MTV4], cf. also [MTV2] and Appendix A in [MTV3].

3.4 The simple joint spectrum of the Bethe subalgebra

It is proved in [R], that for any tensor product of irreducible 𝔤𝔩_{M} modules and for generic z, λ the Bethe subalgebra has a simple joint spectrum. We give here a proof of this fact in the special case of the tensor product L_{n}.

Proposition 9. For generic values of λ, the joint spectrum of the Bethe subalgebra G(M, N, z, λ) acting in L_{n}[m] is simple.

Proof. We claim that for generic values of λ, the joint spectrum of the Gaudin Hamiltonians H_{a}(M, N, z, λ), a = 1, …, N, acting in L_{n}[m] is simple. Indeed fix z and consider λ such that λ_{1} ≫ λ_{2} ≫ ⋯ ≫ λ_{M} ≫ 0. Then the eigenvectors of the Gaudin Hamiltonians in L_{n}[m] will have the form v_{1} ⊗ ⋯ ⊗ v_{N} + o(1), where v_{i} ∈ L_{n_{i}}[m^{(i)}] and m = ∑_{i=1}^{N} m^{(i)}. The corresponding eigenvalue of H_{a}(M, N, z, λ) will be ∑_{j=1}^{M} λ_{j}m_{j}^{(a)} + O(1).

The weight spaces L_{n_{i}}^{(M)}[m_{i}] all have dimension at most 1 and therefore the joint spectrum is simple in this asymptotic zone of parameters. Therefore it is simple for generic values of λ. □

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