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Kähler Structures and Weighted Actions on the Complex Torus

Let T be the compact real torus, and TC its complexification. Fix an integral weight α, and consider the α‐weighted T‐action on TC. If ω is a T‐invariant Kähler form on TC, it corresponds to a pre‐quantum line bundle L over TC. Let Hω be the square‐integrable holomorphic sections of L. The weighted T‐action lifts to a unitary T‐representation on the Hilbert space Hω, 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’.