eBook ISBN:  9781470404222 
Product Code:  MEMO/174/821.E 
List Price:  $87.00 
MAA Member Price:  $78.30 
AMS Member Price:  $52.20 
eBook ISBN:  9781470404222 
Product Code:  MEMO/174/821.E 
List Price:  $87.00 
MAA Member Price:  $78.30 
AMS Member Price:  $52.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 174; 2005; 233 ppMSC: 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 \(GL(n,\mathbb R)\).
ReadershipGraduate 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


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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 \(GL(n,\mathbb R)\).
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