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Fonction zêta associée à la série principale sphérique de certains espaces symétriques

1993; Société Mathématique de France; Volume: 26; Issue: 6 Linguagem: Inglês

10.24033/asens.1685

1873-2151

Nicole Bopp, Hubert Rubenthaler,

Mathematics and Applications

In 1972, Godement and Jacquet obtained thé functional équation of thé zêta function of simple algebras.In thé case where thé groundfîeld is C, such zêta functions are associated to each K x K-fmite coefficient of an irreducible admissible représentation n of GL (n, C).In this paper we consider thé case of a family of complex symmetric spaces G/H which are realized as Zariski open subsets of a vector space (in Godement-Jacquets case, thé space GL(/z, C)wGL(n, C)xGL(/î, C)/GL(n, C) is embedded in thé full matrix space M (n, C)).This family of symmetric spaces corresponds to thé prehomogeneous vector spaces of commutative parabolic type.If n is a generic représentation of thé principal spherical séries of G which has a natural H-invariant generalized vector, we defîne ^oe and H-invariant coefficients of K and thé zêta function associated to thèse coefficients.We obtain an explicit functional équation for this zêta function.