Memoirs of the American Mathematical Society 2005; 233 pp; softcover Volume: 174 ISBN10: 0821836234 ISBN13: 9780821836231 List Price: US$87 Individual Members: US$52.20 Institutional Members: US$69.60 Order Code: MEMO/174/821
 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
