Unitary Representations of Unitary Groups

TL;DR

This paper reviews and extends key results on the structure of unitary representations of infinite-dimensional unitary groups, showing their decomposition into irreducibles and triviality in certain cases.

Contribution

It generalizes the Kirillov--Olshanski theorem to inseparable spaces and clarifies the representation theory of full unitary groups and classical symmetric pairs.

Findings

Separable unitary representations of $ ext{U}( ext{H})$ are determined by their restriction to $ ext{U}_ extinfty( ext{H})_0$.

Representations of $ ext{U}_ ext{H}$ decompose into finite tensor products.

All separable representations are trivial for certain classical symmetric pairs.

Abstract

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group $\U(\cH)$ of a real, complex or quaternionic separable Hilbert space and the subgroup $\U_\infty(\cH)$, consisting of those unitary operators $g$ for which $g - \1$ is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component $\U_\infty(\cH)_0$ asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell's Theorem, asserting that the separable unitary representations of $\U(\cH)$, for a separable Hilbert space $\cH$, are uniquely determined by their restriction to $\U_\infty(\cH)_0$. For the $10$ classical infinite rank symmetric pairs $(G,K)$ of non-unitary type, such as $(\GL(\cH),\U(\cH))$, we also show that all separable unitary representations are trivial.