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Kähler Structures and Weighted Actions on the Complex Torus
Author(s) -
Chuah MengKiat
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008668
Subject(s) - mathematics , torus , holomorphic function , complexification , square integrable function , maximal torus , complex torus , pure mathematics , hilbert space , invariant (physics) , multiplicity (mathematics) , unitary state , line bundle , integrable system , mathematical physics , mathematical analysis , fundamental representation , lie algebra , geometry , political science , law , weight
Let T be the compact real torus, and T C its complexification. Fix an integral weight α, and consider the α‐weighted T ‐action on T C . If ω is a T ‐invariant Kähler 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 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’.