Categorical consequences of the microlocal perspective on Arthur packets for p-adic groups

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This NSERC DG application concerns research in the Langlands programme regarding Arthur packets for p-adic groups. The ultimate objective of this research is to explore a specific idea for a categorical form of the local Langlands correspondence. In the early 1990s, David Vogan introduced the idea of using microlocal analysis on a moduli space of Langlands parameters to study Arthur packets for p-adic groups. This idea is parallel to a construction for Real groups developed in conjunction with Jeffrey Adams and Dan Barbasch in their 1992 book The Langlands Classification and Irreducible Characters of Real Groups. Vogan's idea for p-adic groups was presented in his 1993 paper The Local Langlands Correspondence. However, until James Arthur produced a purely local description of Arthur packets for p-adic groups in his 2013 book The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, it was difficult to test Vogan's ideas. They remain open problems to this date. In Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples by Clifton Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu, Memoirs of the American Mathematical Society (in press), my collaborators and I have adapted many of the results of the book by Adams, Barbasch and Vogan from Real groups to p-adic groups. We refer to the packets of admissible representations of p-adic groups described in our memoir as ABV-packets. The main objective of this proposal is to prove that Arthur packets for pure inner forms of classical p-adic groups are ABV-packets and to prove the refinement of this conjecture appearing in our book pertaining to stable distributions and their endoscopic transfers. I have considerable evidence for this conjecture in the form of examples. Together with my research group, I also have a strategy for the proof of these conjectures for pure inner forms of quasi-split classical groups but because we need to make a careful comparison with Arthur's work, our argument proceeds type by type; with GL(n) in hand, we will work our way through the classical groups beginning with SO(2n+1) and its pure inner forms. At the heart of this research is a comparison of two categories: representations of smooth representations with fixed cuspidal support; and equivariant perverse sheaves on the moduli space of certain Langlands parameters. When properly calibrated, these two categories have the same Grothendieck groups. However, we know that these categories are not, in general, equivalent. My recent work suggests a modification to each of these categories that might be used to establish that they are Koszul dual. This idea is the categorical consequence of the microlocal perspective on Arthur packets that appears in the title of this research proposal and is my ultimate objective: using our work on Arthur packets, I expect to shed light on a categorical local Langlands correspondence for p-adic groups.