Representations of the Infinite Unitary Group from Constrained Quantization

Abstract

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.