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 αweighted T-action on T C . If ω is a T-invariant Ka$ hler 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 subrepresentations 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 '.