
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 |

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 |
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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.
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Table of Contents
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Chapters
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Introduction
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1. A class of real prehomogeneous spaces
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2. The orbits of $G$ in $V^+$
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3. The symmetric spaces $G/H$
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4. Integral formulas
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5. Functional equation of the zeta function for type I and II
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6. Functional equation of the zeta function for type III
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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