Kahler Structures and Weighted Actions on the Complex Torus

Let T be the compact real torus, and T_C its complexification.Fix an integral weight , and consider the -weighted T-actionon T_C. If is a T-invariant Kähler form on T_C, it correspondsto a pre-quantum line bundle L over T_C. Let H be the square-integrableholomorphic sections of L. The weighted T-action lifts to aunitary T-representation on the Hilbert space H, and the multiplicityof its irreducible sub-representations is considered. It isshown that this is controlled by the image of the moment map,as well as the principle that ‘quantization commutes withreduction’.