CONTENTS
Foreword
Preface
The Theta Correspondence over R Jeffrey Adams
1. Introduction
2. Fock Model: Complex Lie Algebra
3. Schrodinger Model
4. Fock Model: Real Lie Algebra
5. Duality
6. Compact Dual Pairs
7. Joint Harmonics
8. Induction Principle
9. Examples
References
The Heisenberg Group, SL(3;R), and Rigidity Andreas Cap, Michael G. Cowling, Filippo De Mari, Michael Eastwood and Rupert McCallum
1. Introduction
2. An Example
3. Related Questions in Two Dimensions
4. Proof of Theorem 2.1
5. Final Remarks
References
Pfafflans and Strategies for Integer Choice Games Ron Evans and Nolan Wallach
1. Introduction
2. Strategies for the Multivariate Game
3. Strategies for the Single Variable Game
4. Strategies for Some Constricted Multivariate Games
5. Appendix: Pfa ans Associated with Payo Matrices
References
When is an L-Function Non-Vanishing in Part of the Critical Strip? Stephen Gelbart
Introduction
1. The Classical Method
2. The Rankin-Selberg Generalization of de la Vall ee Poussin
3. An Approach Using Eisenstein Series on SL(2; R)
4. The General Method
References
Cohomological Automorphic Forms on Unitary Groups, II: Period Relations and Values of L-Functions Michael Harris
Introduction
Errors and Misprints in [H4]
0. Preliminary Notation
1. Eisenstein Series on Unitary Similitude Groups
2. The Local Theta Correspondence
Appendix. Generic Calculation of the Unrami ed Correspondence
3. Applications to Special Values of L-Functions
4. Applications to Period Relations
The Inversion Formula and Holomorphic Extension of the Minimal Representation of the Conformal Group Toshiyuki Kobayashi and Gen Mano
Contents
1. Introduction
1.1. Semigroup generated by a differential operator D
1.2. Comparison with the Hermite operator D
1.3. The action of SL(2;R) O(m)
1.4. Minimal representation as hidden symmetry
2. Preliminary Results on the Minimal Representation of O(m + 1; 2)
2.1. Maximal parabolic subgroup of the conformal group
2.2. L2-model of the minimal representation
2.3. K-type decomposition
2.4. Infinitesimal action of the minimal representation
3. Branching Law of +
3.1. Schr odinger model of the minimal representation
3.2. K- nite functions on the forward light cone C+
3.3. Description of in nitesimal generators of sl(2;R)
3.4. Central element Z of kC
3.5. Proof of Proposition 3.2.1
3.6. One parameter holomorphic semigroup (etZ )
4. Radial Part of the Semigroup
4.1. Result of the section
4.2. Upper estimate of the kernel function
4.3. Proof of Theorem 4.1.1 (Case Re t > 0)
4.4. Proof of Theorem 4.1.1 (Case Re t = 0)
4.5. Weber's second exponential integral formula
4.6. Dirac sequence operators
5. Integral Formula for the Semigroup
5.1. Result of the section
5.2. Upper estimates of the kernel function
5.3. Proof of Theorem 5.1.1 (Case Re t > 0)
5.4. Proof of Theorem 5.1.1 (Case Re t = 0)
5.5. Spectra of an O(m)-invariant operator
5.6. Proof of Lemma 5.3.1
5.7. Expansion formulas
6. The Unitary Inversion Operator
6.1. Result of the section
6.2. Inversion and Plancherel formula
6.3. The Hankel transform
6.4. Forward and backward light cones
7. Explicit Actions of the Whole Group on L2(C)
7.1. Bruhat decomposition of O(m + 1; 2)
7.2. Explicit action of the whole group
8. Appendix: Special Functions
8.1. Laguerre polynomials
8.2. Hermite polynomials
8.3. Gegenbauer polynomials
8.4. Spherical harmonics and Gegenbauer polynomials
8.5. Bessel functions
References
Classification des Series Discretes pour Certains Groupes Classiques p-Adiques Colette MEglin
1. Support Cuspidal des S eries Discr etes
2. Morphismes Associes aux Representations Cuspidales de G(n) et Points de Reductibilite des Induites de Cuspidales
3. Classification et Paquet de Langlands
4. Classification a la Langlands des Series Discretes de SO(2n + 1; F)
5. Le cas des Groupes Orthogonaux Impairs non Deployes
References
Some Algebras of Essentially Compact Distributions of a Reductive p-Adic Group Allen Moy and Marko Tadi c
1. Introduction
2. The Convolution Algebras H(G)b and U(G)
3. Some Properties of the Convolution Algebras H(G)b and U(G)
4. Some Explicit G-invariant Essentially Compact Distributions
Acknowledgment
References
Annihilators of Generalized Verma Modules of the Scalar Type for Classical Lie Algebras Toshio Oshima
1. Introduction
2. Minimal Polynomials
3. Projection to the Cartan Subalgebra
4. Generalized Verma Modules
5. Integral Transforms on Generalized Flag Manifolds
6. Closure of Ideals
References
Branching to a Maximal Compact Subgroup David A. Vogan, Jr.
Contents
1. General Introduction
2. Technical Introduction
3. Highest Weights for K
4. The R-Group of K and Irreducible Representations of the Large Cartan
5. Fundamental Series, Limits, and Continuations
6. Characters of Compact Tori
7. Split Tori and Representations of K
8. Parametrizing Extended Weights
9. Proof of Lemma 8.13
10. Highest Weights for K and -Stable Parabolic Subalgebras
11. Standard Representations, Limits, and Continuations
12. More Constructions of Standard Representations
13. From Highest Weights to Discrete Final Limit Parameters
14. Algorithm for Projecting a Weight on the Dominant Weyl Chamber
15. Making a List of Representations of K
16. G-Spherical Representations as Sums of Standard Representations
References
Small Semisimple Subalgebras of Semisimple Lie Algebras Jeb F. Willenbring and Gregg J. Zuckerman
1. Introduction
2. Invariant Theory
3. Representation Theory
4. A Proof of a Theorem in Penkov-Zuckerman
5. Example: G2
6. Tables for Maximal Parabolic Subalgebras
References