New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
Nicole Bopp and Hubert Rubenthaler, University of Strasbourg, France
 SEARCH THIS BOOK:
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$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 $$GL(n,\mathbb R)$$.

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

• 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