Decomposition spectrale et representations speciales d'un groupe reductif p-adique

Abstract

Let G be a reductive connected p-adic group. With help of the Fourier inversion formula used in [Une formule de Plancherel pour l'algebre de Hecke d'un groupe reductif p-adique - V. Heiermann, Comm. Math. Helv. 76, 388-415, 2001] we give a spectral decomposition on G. In particular we deduce from it essentially that a cuspidal representation of a Levi subgroup M is in the cuspidal support of a square integrable representation of G, if and only if it is a pole of Harish-Chandra's \mu-function of order equal to the parabolic rank of M. This result has been conjectured by A. Silberger in 1978. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra's \mu-function which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coeffieicients. These identities contain informations on the formel degree of square integrable representations and on their position in the induced representation.