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 Uspekhi Mat. Nauk, 2016, Volume 71, Issue 2(428), Pages 121–178 (Mi umn9713)

Endomorphisms of spaces of virtual vectors fixed by a discrete group

F. Rădulescuab

a Università degli Studi di Roma "Tor Vergata", Roma, Italy
b Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, Romania

Abstract: A study is made of unitary representations $\pi$ of a discrete group $G$ that are of type II when restricted to an almost-normal subgroup $\Gamma\subseteq G$. The associated unitary representation $\overline{\pi}^{ p}$ of $G$ on the Hilbert space of ‘virtual’ $\Gamma_0$-invariant vectors is investigated, where $\Gamma_0$ runs over a suitable class of finite-index subgroups of $\Gamma$. The unitary representation $\overline{\pi}^{ p}$ of $G$ is uniquely determined by the requirement that the Hecke operators for all $\Gamma_0$ are the ‘block-matrix coefficients’ of $\overline{\pi}^{ p}$. If $\pi|^ _\Gamma$ is an integer multiple of the regular representation, then there is a subspace $L$ of the Hilbert space of $\pi$ that acts as a fundamental domain for $\Gamma$. In this case the space of $\Gamma$-invariant vectors is identified with $L$. When $\pi|^ _\Gamma$ is not an integer multiple of the regular representation (for example, if $G=\operatorname{PGL}(2,\mathbb Z[1/p])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $\operatorname{PSL}(2,\mathbb R)$, and the $\Gamma$-invariant vectors are cusp forms), $\pi$ is assumed to be the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically, the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of $\Gamma$-invariant vectors. It is proved that the character of the unitary representation $\overline{\pi}^{ p}$ is uniquely determined by the character of the representation $\pi$.
Bibliography: 53 titles.

Keywords: unitary representations, Hecke operators, trace formulae.

 Funding Agency Grant Number Ministero dell'Istruzione, dell'Università e della Ricerca Ministerul Educaţiei şi Cercetării Ştiinţifice PN-II-ID-PCE-2012-4-0201 Supported in part by PRIN-MIUR and by a~grant from the Romanian National Authority for Scientific Research (project no. PN-II-ID-PCE-2012-4-0201).

DOI: https://doi.org/10.4213/rm9713

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English version:
Russian Mathematical Surveys, 2016, 71:2, 291–343

Bibliographic databases:

UDC: 512+517.98
MSC: 11F25, 11F72, 46L65

Citation: F. Rădulescu, “Endomorphisms of spaces of virtual vectors fixed by a discrete group”, Uspekhi Mat. Nauk, 71:2(428) (2016), 121–178; Russian Math. Surveys, 71:2 (2016), 291–343

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9713
• https://doi.org/10.4213/rm9713
• http://mi.mathnet.ru/eng/umn/v71/i2/p121

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Dudko A., Grigorchuk R., “On Irreducibility and Disjointness of Koopman and Quasi-Regular Representations of Weakly Branch Groups”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 51+
2. A. A. Popa, “On the trace formula for Hecke operators on congruence subgroups”, Proc. Amer. Math. Soc., 146:7 (2018), 2749–2764
3. A. A. Popa, “On the trace formula for Hecke operators on congruence subgroups, II”, Res. Math. Sci., 5 (2018), 3, 24 pp.
4. F. Radulescu, “The operator algebra content of the ramanujan-petersson problem”, J. Noncommutative Geom., 13:3 (2019), 805–855
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