Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Unitary Representations of Unitary Groups (1308.1500v1)

Published 7 Aug 2013 in math.RT

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.

Summary

We haven't generated a summary for this paper yet.