Article Representations of the Infinite Unitary Group from Constrained Quantization

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U 0 (H) of all unitary operators U on a separable Hilbert space H for which U ? I is compact, originally found by Kirillov and Ol'shanskii, through constrained quantization of its coadjoint orbits. For this purpose the coadjoint orbits are realized as Marsden-Weinstein quotients. The unconstrained system, given as a Weinstein dual pair, is quantized by a corresponding Howe dual pair. Constrained quantization is then performed in replacing the classical procedure of symplectic reduction by the C-algebraic method of Rieffel induction. Reduction and induction have to be performed with respect to either U (M), which is straightforward, or U (M; N). In the latter case one induces from holomorphic discrete series representations, and the desired result is obtained if one ignores half-forms, and induces from a representation, `half' of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when H is finite-dimensional.