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Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
Nicole Bopp and Hubert Rubenthaler, University of Strasbourg, France
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Memoirs of the American Mathematical Society
2005; 233 pp; softcover
Volume: 174
ISBN-10: 0-8218-3623-4
ISBN-13: 978-0-8218-3623-1
List Price: US$82
Individual Members: US$49.20
Institutional Members: US$65.60
Order Code: MEMO/174/821
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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
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