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Journal of the London Mathematical Society 2000 61(3):937-949; doi:10.1112/S0024610700008668
© 2000 by London Mathematical Society
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© The London Mathematical Society

Kähler Structures and Weighted Actions on the Complex Torus

Meng-Kiat Chuah

Department of Applied Mathematics, National Chiao Tung University 1001 Ta Hsueh Road, Hsinchu, Taiwan, chuah{at}math.nctu.edu.tw

Received 20 May 1998.

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


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