Arthur packets for 𝑝-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples
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Published:2022-03
Issue:1353
Volume:276
Page:
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ISSN:0065-9266
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Container-title:Memoirs of the American Mathematical Society
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language:en
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Short-container-title:Memoirs of the AMS
Author:
Cunningham Clifton,Fiori Andrew,Moussaoui Ahmed,Mracek James,Xu Bin
Abstract
In this article we propose a geometric description of Arthur packets for
p
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
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
p
-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
Publisher
American Mathematical Society (AMS)
Subject
Applied Mathematics,General Mathematics
Reference45 articles.
1. Progress in Mathematics;Adams, Jeffrey,1992
2. Supercuspidal characters of reductive 𝑝-adic groups;Adler, Jeffrey D.;Amer. J. Math.,2009
3. Unipotent automorphic representations: conjectures;Arthur, James;Ast\'{e}risque,1989
4. On the transfer of distributions: weighted orbital integrals;Arthur, James;Duke Math. J.,1999
Cited by
2 articles.
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