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
Forum Mathematicum – de Gruyter
Published: Sep 6, 2000
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