Representations of reductive groups : conference on representation theory and algebraic geometry in honor of Joseph Bernstein, June 11-16, 2017, The Weizmann Institute of Science & The Hebrew University of Jerusalem, Jerusalem, Israel / Avraham Aizenbud [and three others], editors.

Format:

Book

Author/Creator:

Aizenbud, Avraham, 1983- editor.

Series:

Proceedings of symposia in pure mathematics ; Volume 101.

Proceedings of symposia in pure mathematics ; Volume 101

Language:

English

Subjects (All):

Representations of groups--Congresses.

Representations of groups.

Representations of algebras--Congresses.

Representations of algebras.

Physical Description:

1 online resource (466 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2019]

Language Note:

English

Summary:

This volume contains the proceedings of the Conference on Representation Theory and Algebraic Geometry, held in honor of Joseph Bernstein, from June 11-16, 2017, at the Weizmann Institute of Science and The Hebrew University of Jerusalem. The topics reflect the decisive and diverse impact of Bernstein on representation theory in its broadest scope. The themes include representations of p-adic groups and Hecke algebras in all characteristics, representations of real groups and supergroups, theta correspondence, and automorphic forms.

Contents:

Cover

Title page

Contents

Preface

Character values and Hochschild homology

1. Introduction

2. Weightless functions and invariant distributions

2.1. The conjecture

2.2. Almost elliptic orbits

2.3. The case of (2)

3. The compactified category of smooth modules

3.1. Definition of the compactified category

3.2. Compactified center and a spectral description of the compactified category

3.3. The spectral description of \Smb

3.4. Compactified category and filtered modules

4. Hochschild homology and character values

5. (2) calculations

5.1. Explicit complexes for Hochschild homology

5.2. Calculation of ₀

5.3. Cocycles for Chern character and Euler characteristic

5.4. Tate cocycle and weighted orbital integral

5.5. Proof of part (b) of Theorem 5.1

References

Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

1. Introduction and statement of the results

2. Tempered representations and Harish-Chandra algebra

3. Paley-Wiener theorems and the definition of the algebra \calJ( )

4. Intertwining operators

5. Intertwining operators in the cuspidal corank 1 case

6. Proof of \reft{inter-schwartz}

7. Some further questions

Explicit local Jacquet-Langlands correspondence: The non-dyadic wild case

Introduction

1. The Lagrangian subgroup

2. Extensions of simple characters

3. Change of group and endo-classes

4. Transfer via completion

5. The basic character relation

6. Consequences

On the Casselman-Jacquet functor

0.1. The Casselman-Jacquet functor

0.2. Functorial interpretation of the Casselman-Jacquet functor

0.3. The pseudo-identity functor and the ULA condition

0.4. The "2nd adjointness" conjecture

0.5. Organization of the paper.

0.6. Conventions and notation

0.7. How to get rid of DG categories?

0.8. Acknowledgements

1. Recollections

1.1. Groups acting on categories: a reminder

1.2. Localization theory

1.3. Translation functors

1.4. The long intertwining operator

2. Casselman-Jacquet functor as averaging

2.1. Casselman-Jacquet functor in the abstract setting

2.2. Casselman-Jacquet functor as completion

2.3. The Casselman-Jacquet functor for -modules

2.4. The Casselman-Jacquet functor for \fg-modules

2.5. ULA vs finite-generation

3. The pseudo-identity functor

3.1. The pseudo-identity functor: recollections

3.2. Pseudo-identity, averaging and the ULA property

3.3. A variant

3.4. First applications

3.5. Transversality and the proof of \propref{p:equiv ULA}

4. The case of a symmetric pair

4.1. Adjusting the previous framework

4.2. The Casselman-Jacquet functor for (\fg, )-modules

4.3. Proof of \thmref{t:J exact on K}

4.4. The "2nd adjointness" conjecture

Periods and theta correspondence

2. Shalika and Linear Periods

3. Type II Dual Pairs

4. Big Theta Lift

5. A Theorem of Tunnell and Saito

Generalized and degenerate Whittaker quotients and Fourier coefficients

Acknowledgments

1.1. Notation

2. Degenerate Whittaker models and Fourier coefficients

2.1. Definitions

2.2. Fourier coefficients

2.3. Comparison between different Whittaker pairs

2.4. The Slodowy slice

3. Admissible and quasi-admissible orbits

3.1. Definitions

3.2. On the proof of Theorem B

4. Wave-front sets

4.1. Definition

4.2. On the proof of Theorem A

4.3. Archimedean case

Geometric approach to the fermionic Fock space, via flag varieties and representations of algebraic (super)groups

Introduction.

1. Basic setting

2. Fock space

3. Duality between geometric induction and restriction

4. Two functors on ℱ

5. Link with the Clifford algebra

6. Translation functors

Representations of a -adic group in characteristic

I. Introduction

II. Some general algebra

II.1. Review on scalar extension

II.2. A bit of ring and module theory

II.3. Proof of the decomposition theorem (Thm.I.1 and Cor.I.2)

II.4. Proof of the lattice theorems (Thm. I.3, I.5 and Cor. I.4, I.6)

III. Classification theorem for

III.1. Admissibility, -invariants, and scalar extension

III.2. Decomposition Theorem for

III.3. The representations _{ }( , , )

III.4. Supersingular representations

III.5. Classification of irreducible admissible -representations of

IV. Classification theorem for ( )

IV.1. Pro- Iwahori Hecke ring

IV.2. Parabolic induction \Ind_{ }^{ ( )}

IV.3. The ( )_{ }-module \St_{ }^{ ( )}(\cV)

IV.4. The module _{ ( )}( ,\cV, )

IV.5. Classification of simple modules over the pro- Iwahori Hecke algebra

V. Applications

V.1. Vanishing of the smooth dual

V.2. Lattice of submodules (Proof of Theorem I.10)

V.3. Proof of Theorem I.12

VI. Appendix: Eight inductions \Mod_{ }( ( ))→\Mod_{ }( ( ))

On the support of matrix coefficients of supercuspidal representations of the general linear group over a local non-archimedean field

2. A variant for Whittaker functions

3. Proof of main result

On the generalized Springer correspondence

1. Preliminaries

2. A trace computation

3. Computations in certain groups of semisimple rank \le5

4. Euler characteristic computations

5. The main result

6. Final comments

The modular pro- Iwahori-Hecke -algebra

1. Introduction.

2. Notations and preliminaries

2.1. Elements of Bruhat-Tits theory

2.2. The pro- Iwahori-Hecke algebra

2.3. Supersingularity

3. The \Ext-algebra

3.1. The definition

3.2. The technique

3.3. The cup product

4. Representing cohomological operations on resolutions

4.1. The Shapiro isomorphism

4.2. The Yoneda product

4.3. The cup product

4.4. Conjugation

4.5. The corestriction

4.6. Basic properties

5. The product in *

5.1. A technical formula relating the Yoneda and cup products

5.2. Explicit left action of on the Ext-algebra

5.3. Appendix

6. An involutive anti-automorphism of the algebra *

7. Dualities

7.1. Finite and twisted duals

7.2. Duality between ⁱ and ^{ - } when is a Poincaré group

8. The structure of ^{ }

Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

2. The conjecture of Hiraga, Ichino and Ikeda

2.1. The decomposition of the trace

2.2. Normalization of Haar measure

2.3. Local Langlands parameters

2.4. -functions and factors

2.5. A conjectural tempered local Langlands correspondence

2.6. The conjectures of Hiraga, Ichino and Ikeda

2.7. Known results and further comments

3. The Plancherel formula for affine Hecke algebras

3.1. The Bernstein center

3.2. Types, Hecke algebras and Plancherel measure

3.3. Affine Hecke algebras as Hilbert algebras

3.4. A formula for the trace of an affine Hecke algebra

3.5. Spectral decomposition of

3.6. Residual cosets and their properties

3.7. Deformation of discrete series and the computation of _{\Hc, }

3.8. Central characters and Langlands parameters

4. Lusztig's representations of unipotent reduction and spectral transfer maps. Main result.

4.1. Unipotent types and unipotent affine Hecke algebras

4.2. Langlands parameters and residual cosets

4.3. Spectral transfer maps

4.4. Lusztig's geometric-arithmetic correspondences and STMs

4.5. Main Theorem

Limiting cycles and periods of Maass forms

2. Proof

On vector-valued twisted conjugation invariant functions on a group

with an appendix by Stephen Donkin

2. Filtered vector spaces and Rees modules

3. Filtration on representations

4. Vector-valued twisted conjugation invariant functions

5. Chevalley groups with an automorphism

6. The determinant of the pairing \bfJ( )⊗\bfJ( *)→\bfJ

Appendix: A remark on freeness

Local theta correspondence and nilpotent invariants

2. Dual pairs and local theta correspondence: a brief review

3. Correspondence of generalized Whittaker models

4. Correspondence of associated characters

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-5157-3

OCLC:

1089985402