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2014 | OriginalPaper | Buchkapitel

Unitary Representations of Unitary Groups

verfasst von : Karl-Hermann Neeb

Erschienen in: Developments and Retrospectives in Lie Theory

Verlag: Springer International Publishing

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Abstract

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\) of a real, complex or quaternionic separable Hilbert space and the subgroup \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})\), 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 \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\) 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 \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\), for a separable Hilbert space \(\mathcal{H}\), are uniquely determined by their restriction to \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\). For the 10 classical infinite rank symmetric pairs (G, K) of non-unitary type, such as \((\mathop{\mathrm{GL}}\nolimits (\mathcal{H}),\mathop{\mathrm{U}}\nolimits (\mathcal{H}))\), we also show that all separable unitary representations are trivial.

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Vorheriges Kapitel Twisted Harish–Chandra Sheaves and Whittaker Modules: The Nondegenerate Case

Nächstes Kapitel Weak Splittings of Quotients of Drinfeld and Heisenberg Doubles

Nur mit Berechtigung zugänglich

Actually this group is connected for \(\mathbb{K} = \mathbb{C},\mathbb{H}\) [Ne02, Cor. II.15].

For \(\mathbb{K} = \mathbb{C},\mathbb{H}\) , the condition \(2\mathop{ \mathrm{dim}}\nolimits \mathcal{F}\leq \mathop{\mathrm{dim}}\nolimits \mathcal{H}\) is sufficient.

Our assumption implies that \(\mathop{\mathrm{dim}}\nolimits \mathcal{H}\geq 2\). This claim follows from the case where \(\mathcal{H} = \mathbb{K}^{2}\). Using the diagonal inclusion \(\mathop{\mathrm{U}}\nolimits (1,\mathbb{K})^{2}\hookrightarrow \mathop{ \mathrm{U}}\nolimits (2,\mathbb{K})\), it suffices to consider vectors with real entries, which reduces the problem to the transitivity of the action of \(\mathop{\mathrm{SO}}\nolimits (2,\mathbb{R})\) on the unit circle. Since the trivial group \(\mathop{\mathrm{SO}}\nolimits (1,\mathbb{R})\) does not act transitively on \(\mathbb{S}^{0} =\{ \pm 1\}\), it is here where we need that \(2\mathop{ \mathrm{dim}}\nolimits \mathcal{F} <\mathop{ \mathrm{dim}}\nolimits \mathcal{H}\).

This follows from the fact that \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\) acts transitively on the finite orthonormal systems in \(\mathcal{H}\).

This argument simplifies Pickrell’s argument that was based on the simplicity of the topological group \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})/\mathbb{T}\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})\) [Ka52].

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Titel: Unitary Representations of Unitary Groups
verfasst von: Karl-Hermann Neeb
Verlag: Springer International Publishing
Buch: Developments and Retrospectives in Lie Theory
Print ISBN: 978-3-319-09933-0

Electronic ISBN: 978-3-319-09934-7

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-09934-7_8

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