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Zbl 0806.14039
Bopp, Nicole; Rubenthaler, Hubert
Zeta function associated to the spherical principal series of some symmetric spaces. (Fonction zêta associée à la série principale sphérique de certains espaces symétriques.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 26, No. 6, 701-745 (1993). ISSN 0012-9593

The theme of this paper is a trial of generalization of the work ``Zeta functions of simple algebras'' by {\it R. Godement} and {\it H. Jacquet} [Lect. Notes Math. 260 (1972; Zbl 0244.12011)] from a viewpoint of the theory of prehomogeneous vector spaces. The authors consider the case of a family of complex symmetric spaces $G/H$ which are obtained as Zariski open subsets of a vector space. The family of symmetric spaces corresponds to the prehomogeneous vector spaces of commutative parabolic type. For example, in Godement and Jacquet's case, they considered $GL\sb n (\bbfC) \simeq GL\sb n (\bbfC) \times GL\sb n (\bbfC)/GL\sb n (\bbfC)$. It is embedded in the space of $n \times n$ complex matrices $M\sb n (\bbfC)$. The action $GL\sb n (\bbfC) \times GL\sb n (\bbfC)$ on $M\sb n (\bbfC)$ is a typical example of prehomogeneous vector space of commutative parabolic type. If $\pi$ is a a generic representation of the principal spherical series of $G$ which has a natural $H$-invariant generalized vector, we define $C\sp \infty$ and $H$-invariant coefficients of $\pi$ and the zeta function associated to these coefficients. We obtain an explicit functional equation for this zeta function.
[M.Muro (Yanagido)]

MSC 2000:: *14M17 Homogeneous spaces
14G10 Zeta-functions and related questions
32M15 Symmetric spaces (analytic spaces)

Keywords: symmetric spaces; prehomogeneous vector space of commutative parabolic type; zeta function

Citations: Zbl 0244.12011

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