Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples / Clifton L.R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, Bin Xu.

Format:

Book

Author/Creator:

Cunningham, Clifton, 1966-

Contributor:

Fiori, Andrew.

Moussaoui, Ahmed.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

p-adic groups.

Representations of groups.

Sheaf theory.

Number theory.

Physical Description:

1 online resource (232 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"In this article we propose a geometric description of Arthur packets for padic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, including the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples"-- Provided by publisher.

Contents:

Cover

Title page

Acknowledgments

Chapter 1. Introduction

1.1. Motivation

1.2. Background

1.3. Main results

1.4. Conjecture

1.5. Examples

1.6. Relation to other work

1.7. Disclaimer

Part 1. Arthur packets and microlocal vanishing cycles

Chapter 2. Overview

Chapter 3. Arthur packets and pure rational forms

3.1. Local Langlands group

3.2. L-groups

3.3. Semisimple, elliptic and hyperbolic elements in L-groups

3.4. Langlands parameters

3.5. Arthur parameters

3.6. Langlands parameters of Arthur type

3.7. Pure rational forms

3.8. Langlands packets for pure rational forms

3.9. Arthur packets for quasisplit symplectic or special orthogonal groups

3.10. Arthur packets for inner rational forms

3.11. Pure Arthur packets

3.12. Virtual representations of pure rational forms

Chapter 4. Equivariant perverse sheaves on parameter varieties

4.1. Infinitesimal parameters

4.2. Vogan varieties

4.3. Parameter varieties

4.4. Equivariant perverse sheaves

4.5. Equivariant perverse sheaves on parameter varieties

4.6. Langlands component groups as equivariant fundamental groups

Chapter 5. Reduction to unramified hyperbolic parameters

5.1. Unramification

5.2. Elliptic and hyperbolic parts of the image of Frobenius

5.3. Construction of the unramified parameter

5.4. Construction of the cocharacter

5.5. Proof of Theorem 5.1

5.6. Further properties of Vogan varieties

Chapter 6. Arthur parameters and the conormal bundle

6.1. Regular conormal vectors

6.2. Cotangent space to the Vogan variety

6.3. Conormal bundle to the Vogan variety

6.4. Orbit duality

6.5. Strongly regular conormal vectors

6.6. From Arthur parameters to strongly regular conormal vectors

6.7. Arthur component groups as equivariant fundamental groups.

6.8. Proof of Proposition 6.1

6.9. Equivariant Local systems

Chapter 7. Microlocal vanishing cycles of perverse sheaves

7.1. Background on vanishing cycles

7.2. Calculating vanishing cycles

7.3. Brylinski's functor Ev

7.4. Stalks

7.5. Support

7.6. Open orbit

7.7. Purity and rank on the open orbit

7.8. Perversity

7.9. Restriction to generic elements

7.10. Normalization of Ev

7.11. Remarks on stratified Morse theory and microlocalization

Chapter 8. Arthur packets and ABV-packets

8.1. ABV-packets

8.2. Virtual representations attached to ABV-packets

8.3. Conjecture on Arthur packets

8.4. A basis for stable invariant distributions

Part 2. Examples

Chapter 9. Overview

Chapter 10. Template for the examples

10.1. Arthur packets

10.2. Vanishing cycles of perverse sheaves

10.3. ABV-packets

10.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 11. SL(2) 4-packet of quadratic unipotent representations

11.1. Arthur packets

11.2. Vanishing cycles of perverse sheaves

11.3. ABV-packets

11.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 12. SO(3) unipotent representations, regular parameter

12.1. Arthur packets

12.2. Vanishing cycles of perverse sheaves

12.3. ABV-packets

12.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 13. PGL(4) shallow representations

13.1. Arthur packets

13.2. Vanishing cycles of perverse sheaves

13.3. ABV-packets

13.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 14. SO(5) unipotent representations, regular parameter

14.1. Arthur packets

14.2. Vanishing cycles of perverse sheaves

14.3. ABV-packets

14.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 15. SO(5) unipotent representations, singular parameter.

15.1. Arthur packets

15.2. Vanishing cycles of perverse sheaves

15.3. ABV-packets

15.4. Endoscopy and equivariant restriction of perverse sheaves

Chapter 16. SO(7) unipotent representations, singular parameter

16.1. Arthur packets

16.2. Vanishing cycles of perverse sheaves

16.3. ABV-packets

16.4. Endoscopy and equivariant restriction of perverse sheaves

16.5. Tables for the SO(7) example

Bibliography

Index

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Cunningham, Clifton Arthur Packets for p-adic Groups by Way of Microlocal Vanishing Cycles of Perverse Sheaves, with Examples

ISBN:

9781470470197

1470470195

OCLC:

1309012471