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Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
 
Nicole Bopp University of Strasbourg, Strasbourg, France
Hubert Rubenthaler University of Strasbourg, Strasbourg, France
Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
eBook ISBN:  978-1-4704-0422-2
Product Code:  MEMO/174/821.E
List Price: $87.00
MAA Member Price: $78.30
AMS Member Price: $52.20
Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
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Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
Nicole Bopp University of Strasbourg, Strasbourg, France
Hubert Rubenthaler University of Strasbourg, Strasbourg, France
eBook ISBN:  978-1-4704-0422-2
Product Code:  MEMO/174/821.E
List Price: $87.00
MAA Member Price: $78.30
AMS Member Price: $52.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1742005; 233 pp
    MSC: Primary 17; 22; 32; 43

    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
     
     
    • Chapters
    • Introduction
    • 1. A class of real prehomogeneous spaces
    • 2. The orbits of $G$ in $V^+$
    • 3. The symmetric spaces $G/H$
    • 4. Integral formulas
    • 5. Functional equation of the zeta function for type I and II
    • 6. Functional equation of the zeta function for type III
    • 7. Zeta function attached to a representation in the minimal spherical principal series
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1742005; 233 pp
MSC: Primary 17; 22; 32; 43

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.

  • Chapters
  • Introduction
  • 1. A class of real prehomogeneous spaces
  • 2. The orbits of $G$ in $V^+$
  • 3. The symmetric spaces $G/H$
  • 4. Integral formulas
  • 5. Functional equation of the zeta function for type I and II
  • 6. Functional equation of the zeta function for type III
  • 7. Zeta function attached to a representation in the minimal spherical principal series
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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