Abstract
Consider the complex torus T C under the natural action of the compact real torus T. In this paper, we study T-invariant Kahler structures ω on T C . For each ω, we consider the corresponding line bundle L on T C . Namely, the Chem class of L is [ω], and L is equipped with a connection V whose curvature is ω. We construct a canonical T-invariant L 2 -structure on the sections of L, and let H ω , be the square-integrable holomorphic sections of L. Then the Hilbert space Hω is a unitary T-representation, and we study the multiplicity of the (1-dimensional) irreducible unitary T-representations in Hω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω. © 2000 The Journal of Geometric Analysis.