Representations of the infinite unitary group from constrained quantization

TL;DR

This paper explores reconstructing irreducible unitary representations of a specific Banach Lie group using constrained quantization of coadjoint orbits, employing symplectic reduction and Rieffel induction techniques.

Contribution

It introduces a novel constrained quantization approach for infinite-dimensional unitary groups via symplectic reduction and $C^*$-algebraic methods, extending representation theory.

Findings

Successfully reconstructs representations for finite-dimensional cases.

Demonstrates the use of Rieffel induction in constrained quantization.

Highlights limitations in infinite-dimensional cases due to half-form considerations.

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 $U-{\mathbb I}$ 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 \rep, `half' of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when $\H$ is finite-dimensional.