Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces (Memoirs of the American Mathematical Society No. 174)

Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces (Memoirs of the American Mathematical Society No. 174)

By: Nicole Bopp (author), Hubert Rubenthaler (author)Paperback

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Description

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 $G\slash H_p$.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 $GL(n,\mathbb R)$.

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Contents

Introduction A class of real prehomogeneous spaces The orbits of $G$ in $V^+$ The symmetric spaces $G\slash H$ 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.

Product Details

  • publication date: 15/01/2005
  • ISBN13: 9780821836231
  • Format: Paperback
  • Number Of Pages: 233
  • ID: 9780821836231
  • weight: 454
  • ISBN10: 0821836234

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