A formula for the multiplicitiy of the principal series in L 2 (क\ G)

A formula for the multiplicitiy of the principal series in L 2 (क\ G) Abstract. The purpose of this paper is to give a multiplicity formula of the principal series representation which appears in the decomposition of the regular representation on q nG as a certain limit of the integral representation in terms of the logarithmic derivative of the Selberg zeta function. 1991 Mathematics Subject Classi®cation: 11F72, 11M41, 22E40. 0 Introduction It is quite interesting to know the explicit formula of the multiplicity of each irreducible unitary representation in the decomposition of the regular representation of the non-compact Lie group G on a co-®nite, or a co-compact quotient of a negatively curved Riemannian symmetric space by a discrete subgroup. It is, of course, wellknown that the multiplicity of the unitary principal series is described as the order of a zero of the Selberg zeta function (see [G], [GW], [W]). In spite of this fact, if one wants to know the multiplicity as a very explicit quantity, and/or rather to ®nd (uniform) bounds (if any) of multiplicities, it would be useful if there is another expression of the multiplicity. On the other hand, as to the case of the discrete series representations of G, it is also well-known that the multiplicity of a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

A formula for the multiplicitiy of the principal series in L 2 (क\ G)

Forum Mathematicum, Volume 12 (6) – Sep 6, 2000

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Publisher
de Gruyter
Copyright
Copyright © 2000 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2000.027
Publisher site
See Article on Publisher Site

Abstract

Abstract. The purpose of this paper is to give a multiplicity formula of the principal series representation which appears in the decomposition of the regular representation on q nG as a certain limit of the integral representation in terms of the logarithmic derivative of the Selberg zeta function. 1991 Mathematics Subject Classi®cation: 11F72, 11M41, 22E40. 0 Introduction It is quite interesting to know the explicit formula of the multiplicity of each irreducible unitary representation in the decomposition of the regular representation of the non-compact Lie group G on a co-®nite, or a co-compact quotient of a negatively curved Riemannian symmetric space by a discrete subgroup. It is, of course, wellknown that the multiplicity of the unitary principal series is described as the order of a zero of the Selberg zeta function (see [G], [GW], [W]). In spite of this fact, if one wants to know the multiplicity as a very explicit quantity, and/or rather to ®nd (uniform) bounds (if any) of multiplicities, it would be useful if there is another expression of the multiplicity. On the other hand, as to the case of the discrete series representations of G, it is also well-known that the multiplicity of a

Journal

Forum Mathematicumde Gruyter

Published: Sep 6, 2000

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