**Memoirs of the American Mathematical Society**

2005;
233 pp;
Softcover

MSC: Primary 17; 22; 32; 43;

Print ISBN: 978-0-8218-3623-1

Product Code: MEMO/174/821

List Price: $87.00

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MAA Member Price: $78.30

**Electronic ISBN: 978-1-4704-0422-2
Product Code: MEMO/174/821.E**

List Price: $87.00

AMS Member Price: $52.20

MAA Member Price: $78.30

# Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces

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*Nicole Bopp; Hubert Rubenthaler*

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

# Table of Contents

## Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces

- Table of Contents v6 free
- Introduction 110 free
- Chapter 1. A Class of Real Prehomogeneous Spaces 918 free
- 1.1. A class of graded algebras 918
- 1.2. Root systems 1019
- 1.3. Complexification 1120
- 1.4. Highest root in Σ 1221
- 1.5. The first step for the descent 1322
- 1.6. The descent 1625
- 1.7. Generic elements in V[sup(+)] 1726
- 1.8. Structure of the regular graded algebra (g, H[sub(0)]) 2029
- 1.9. Properties of the spaces E[sub(i,j)] (p, q) 2433
- 1.10. Normalization of the Killing form 2635
- 1.11. The relative invariant Δ[sub(0)] 2736
- 1.12. The case k = 0 2837
- 1.13. Properties of Δ[sub(0)] 3140
- 1.14. The polynomials Δ[sub(j)] 3342

- Chapter 2. The Orbits of G in V[sup(+)] 3746
- 2.1. Representations of sl( 2, C) 3746
- 2.2. First reduction 3847
- 2.3. An involution which permutes the roots in E[sub(i,j)(+1,+1) 4049
- 2.4. Construction of elements interchanging λ[sub(i)] and λ[sub(j)] 4150
- 2.5. Quadratic forms 4453
- 2.6. The G-orbits for Type III 4554
- 2.7. The G-orbits for Type II 4655
- 2.8. Signature of the quadratic forms qx[sub(i)],x[sub(j)] 4857
- 2.9. Action of Z[sub(G)](I[sup(+)]) for Type I 5261
- 2.10. The G–orbits for Type I 5463
- 2.11. The classification 5665

- Chapter 3. The Symmetric Spaces G / H 5968
- 3.1. The involutions 5968
- 3.2. In Type I case θ is one of the σ's 6271
- 3.3. Root systems and Types I, II and III 6473
- 3.4. The groups K, H and G[sup(σ)] 6776
- 3.5. A parabolic subgroup of G 7180
- 3.6. The prehomogeneous vector space ( P, V[sup(+)]) 7584
- 3.7. The involution γof g 7887
- 3.8. The orbits of P in V[sup(-) and the polynomials ∇[sub(j)] 8190
- 3.9. A G…equivariant map from Ω[sup(+)] onto Ω[sup(-)] 8493
- 3.10. Intersection of N–orbits with the diagonal 8695

- Chapter 4. Integral Formulas 91100
- 4.1. Integrals on V[sup(+)] and V[sup(-)] 91100
- 4.2. Two diffeomorphisms 96105
- 4.3. Isomorphisms between g(1), g (–1), V[sup(+)]( l ) and V[sup(-)](–1) 98107
- 4.4. A first normalization and its consequence 100109
- 4.5. A second normalization and its consequence 102111
- 4.6. Integral formulas on V[sup(+)] and V[sup(-)] 103112
- 4.7. Fourier transform of a quadratic character 105114
- 4.8. A relation between T[sup(-)][sub(Ff)] and T[sup(+)][sub(f)] 107116

- Chapter 5. Functional Equation of the Zeta Functionfor Type I and II 111120
- 5.1. Definition of the local Zeta functions 111120
- 5.2. Existence of a functional equation for (AN, V[sup(+)]) 112121
- 5.3. Computation of the coefficients δ(s, ε,n) 119128
- 5.4. Computation of the coefficients υ(s, n,ε) 125134
- 5.5. Functional equation for ( P, V[sup(+)]) 127136
- 5.6. A generalized functional equation for ( P, V[sup(+)]) in Type II 129138
- 5.7. Functional equation for ( G, V[sup(+)]) 132141

- Chapter 6. Functional Equation of the Zeta Functionfor Type III 137146
- 6.1. (M ∩ P)…spherical representations of M 137146
- 6.2. Equivariant mappings from O[sup(+)] into H[sub(τ)] 140149
- 6.3. Definition of the local Zeta functions 144153
- 6.4. Existence of a functional equation for ( P, V[sup(+)]) 145154
- 6.5. Explicit functional equation for k = 0 149158
- 6.6. Computation of the coefficients δ(s, τ) 153162
- 6.7. Computation of the coefficients v(s, τ) 157166

- Chapter 7. Zeta Function Attached to a Representation in the MinimalSpherical Principal Series 159168
- 7.1. The minimal spherical principal series 160169
- 7.2. Summary with unified notations 162171
- 7.3. Differentiable vectors and distribution vectors for the minimal principal spherical series 166175
- 7.4. H[sub(p)]–invariant distribution vectors for the minimal principal spherical series 169178
- 7.5. Another expression of a[sup(p,γ)][sub(τ,λ)]in the continuous case 172181
- 7.6. The Zeta integral of an H–invariant distribution vector 175184
- 7.7. A formal functional equation 177186
- 7.8. Computation of Z[sup(+)][sub(p)](f,π[sub(τ,λa[sup(p,γ)][sub(τ,λ)] 180189
- 7.9. The Zeta function attached to the minimal spherical principal series and its functional equation ( Main Theorem) 183192
- 7.10. Fourier transform on L[sup(2)](V[sup(+)], d* X) 186195
- 7.11. Interpretation of the functional equation with F* 190199

- Appendix: The Example of Symmetric Matrices 193202
- Tables of Simple Regular Graded Lie Algebras 223232
- References 227236
- Index 231240