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 TC. For each ω, we consider the corresponding line bundleL on T C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ∇ whose curvature is ω. We construct a canonical T-invariant L 2-structure on the sections ofL,and let H ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H ω is a unitary T-representation, and we study the multiplicity of the (l-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.
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