Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

Clifton L. R. Cunningham University of Calgary Andrew Fiori University of Lethbridge Ahmed Moussaoui Université de Poitiers James Mracek Amazon, San Francisco Bin Xu Yau Mathematical Science Center, Tsinghua University

Number Theory Representation Theory Algebraic Geometry mathscidoc:2206.24001

Memoirs of the American Mathematical Society, 276, (1353), 2022.3
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, especially the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
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@inproceedings{clifton2022arthur,
  title={Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples},
  author={Clifton L. R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, and Bin Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220618161236538115405},
  booktitle={Memoirs of the American Mathematical Society},
  volume={276},
  number={1353},
  year={2022},
}
Clifton L. R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, and Bin Xu. Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples. 2022. Vol. 276. In Memoirs of the American Mathematical Society. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220618161236538115405.
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