The Ham Sandwich Theorem -- any N finite measures on R^N can
be simultaneously bisected by single hyperplane -- is the most
classical result of equipartition theory, a topic central to geometric
and topological combinatorics. We provide group-theoretic
generalizations of this result, showing how finite measures can be
``G-balanced'' by unitary representations of a compact Lie group
G. For abelian groups, such G-Ham Sandwich Theorems have an
equivalent interpretation in terms of vanishing Fourier transforms. In
the finite cases, these yield (equi-)partitions by families of complex
regular q-fans of varying q, complex analogues of the famous
Gr\"unabum problem on equipartitions by families of hyperplanes (i.e.,
regular 2-fans). For the torus groups T^k, one has center transversal
theorems in an L^2-sense for families of complex hyperplanes, similar
in spirit to the center-point theorem of Rado and its duals.