A study is made of unitary representations π of a discrete group G that are of type II when restricted to an almost-normal subgroup Γ ⊆ G. The associated unitary representation π p of G on the Hilbert space of ‘virtual’ Γ0-invariant vectors is investigated, where Γ0 runs over a suitable class of finite-index subgroups of Γ. The unitary representation π p of G is uniquely determined by the requirement that the Hecke operators for all Γ0 are the ‘block-matrix coefficients’ of π p. If π|Γ is an integer multiple of the regular representation, then there is a subspace L of the Hilbert space of π that acts as a fundamental domain for Γ. In this case the space of Γ-invariant vectors is identified with L. When π|Γ is not an integer multiple of the regular representation (for example, if G = PGL(2, Z[1/p]), Γ is the modular group, π belongs to the discrete series of representations of PSL(2,R), and the Γ-invariant vectors are cusp forms), π is assumed to be the restriction to a subspace H0 of a larger unitary representation having a subspace L as above. The operator angle between the projection PL onto L (typically, the characteristic function of the fundamental domain) and the projection P0 onto the subspace H0 (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of Γ-invariant vectors. It is proved that the character of the unitary representation π p is uniquely determined by the character of the representation π. Bibliography: 53 titles. Keywords: unitary representations, Hecke

Radulescu, F. (2016). Endomorphisms ofspaces ofvirtual vectors fixed by a discrete group. RUSSIAN MATHEMATICAL SURVEYS, 71(2), 291-343 [10.1070/RM9713].

Endomorphisms ofspaces ofvirtual vectors fixed by a discrete group

RADULESCU, FLORIN
2016

Abstract

A study is made of unitary representations π of a discrete group G that are of type II when restricted to an almost-normal subgroup Γ ⊆ G. The associated unitary representation π p of G on the Hilbert space of ‘virtual’ Γ0-invariant vectors is investigated, where Γ0 runs over a suitable class of finite-index subgroups of Γ. The unitary representation π p of G is uniquely determined by the requirement that the Hecke operators for all Γ0 are the ‘block-matrix coefficients’ of π p. If π|Γ is an integer multiple of the regular representation, then there is a subspace L of the Hilbert space of π that acts as a fundamental domain for Γ. In this case the space of Γ-invariant vectors is identified with L. When π|Γ is not an integer multiple of the regular representation (for example, if G = PGL(2, Z[1/p]), Γ is the modular group, π belongs to the discrete series of representations of PSL(2,R), and the Γ-invariant vectors are cusp forms), π is assumed to be the restriction to a subspace H0 of a larger unitary representation having a subspace L as above. The operator angle between the projection PL onto L (typically, the characteristic function of the fundamental domain) and the projection P0 onto the subspace H0 (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of Γ-invariant vectors. It is proved that the character of the unitary representation π p is uniquely determined by the character of the representation π. Bibliography: 53 titles. Keywords: unitary representations, Hecke
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
Reopresentations, Ramanujan Petersson cnjectures
Radulescu, F. (2016). Endomorphisms ofspaces ofvirtual vectors fixed by a discrete group. RUSSIAN MATHEMATICAL SURVEYS, 71(2), 291-343 [10.1070/RM9713].
Radulescu, F
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/171164
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