Kahler 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 alpha, and consider the alpha-weighted T-action on T-C. If omega is a T-invariant Kahler form on T-C, it corresponds to a pre-quantum line bundle L over T-C. Let H-omega be the square-integrable holomorphic sections of L. The weighted T-action lifts to a unitary T-representation on the Hilbert space H-omega, 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'.