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Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples
About this Title
Clifton L. R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 276, Number 1353
ISBNs: 978-1-4704-5117-2 (print); 978-1-4704-7019-7 (online)
DOI: https://doi.org/10.1090/memo/1353
Published electronically: March 9, 2022
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
1. Arthur packets and microlocal vanishing cycles
- 2. Overview
- 3. Arthur packets and pure rational forms
- 4. Equivariant perverse sheaves on parameter varieties
- 5. Reduction to unramified hyperbolic parameters
- 6. Arthur parameters and the conormal bundle
- 7. Microlocal vanishing cycles of perverse sheaves
- 8. Arthur packets and ABV-packets
2. Examples
- 9. Overview
- 10. Template for the examples
- 11. SL(2) 4-packet of quadratic unipotent representations
- 12. SO(3) unipotent representations, regular parameter
- 13. PGL(4) shallow representations
- 14. SO(5) unipotent representations, regular parameter
- 15. SO(5) unipotent representations, singular parameter
- 16. SO(7) unipotent representations, singular parameter
Abstract
In this article we propose a geometric description of Arthur packets for $p$-adic 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.- Jeffrey Adams, Dan Barbasch, and David A. Vogan Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1162533, DOI 10.1007/978-1-4612-0383-4
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