KÄHLER STRUCTURES AND WEIGHTED ACTIONS ON THE COMPLEX TORUS

Abstract

Let T be the compact real torus, and T_C its complexification. Fix an integral weight α, and consider the α-weighted T-action on T_C. If ω is a T-invariant Kähler form on T_C, it corresponds to a pre-quantum line bundle L over T_C. Let H_ω be the square-integrable holomorphic sections of L. The weighted T-action lifts to a unitary T-representation on the Hilbert space H_ω, and the multiplicity of its irreducible sub-representations is considered. It is shown that this is controlled by the image of the moment map, as well as the principle that ‘quantization commutes with reduction’.