The aim of this paper is to prove a functional equation for a local zeta function attached to the minimal spherical series for a class of real reductive symmetric spaces. These symmetric spaces are obtained as follows. We consider a graded simple real Lie algebra \(\widetilde{\mathfrak g}\) of the form \(\widetilde{\mathfrak g}=V^-\oplus \mathfrak g\oplus V^+\), where \([\mathfrak g,V^+]\subset V^+\), \([\mathfrak g,V^-]\subset V^-\) and \([V^-,V^+]\subset \mathfrak g\). If the graded algebra is regular, then a suitable group \(G\) with Lie algebra \(\mathfrak g\) has a finite number of open orbits in \(V^+\), each of them is a realization of a symmetric space \(G / H_p\). The functional equation gives a matrix relation between the local zeta functions associated to \(H_p\)-invariant distributions vectors for the same minimal spherical representation of \(G\). This is a generalization of the functional equation obtained by Godement} and Jacquet for the local zeta function attached to a coefficient of a representation of \(GL(n,\mathbb R)\).

Readership

Graduate students and research mathematicians interested in number theory and representation theory.

Table of Contents

• Introduction
• A class of real prehomogeneous spaces
• The orbits of \(G\) in \(V^+\)
• The symmetric spaces \(G / H\)
• Integral formulas
• Functional equation of the zeta function for Type I and II
• Functional equation of the zeta function for Type III
• Zeta function attached to a representation in the minimal spherical principal series
• Appendix: The example of symmetric matrices
• Tables of simple regular graded Lie algebras
• References
• Index