Mathematics Seminar
Date: Friday, 18 August 2023
Time: 2:00 - 3:00 PM
Venue: Seminar Hall
Geometry of local Arthur packets and Vogan's conjecture for $p$-adic $GL_n$
Mishty Ray
University of Calgary.
18-08-23

---

Abstract

The local Langlands correspondence for a connected reductive $p$-adic group G partitions the set of equivalence classes of smooth irreducible representations of G(F) into L-packets using Langlands parameters. Arthur's work introduces the notion of A-packets, which are sets of unitary irreducible representations of G(F) attached to Arthur parameters. On the other hand, Vogan's geometric perspective on the Langlands correspondence establishes a bijection between equivalence classes of smooth irreducible representations of G(F) (along with its pure inner forms) and simple equivariant perverse sheaves on a moduli space of Langlands parameters. This theory gives us the notion of an ABV-packet, which is also a set of smooth irreducible representations of G(F), but now attached to any Langlands parameter. Conjecturally, ABV-packets are generalized A-packets; we call this Vogan's conjecture. In recent work, joint with Clifton Cunningham, we proved this conjecture for $G=GL_n$. In this talk, we will explore the geometry of the moduli space of Langlands parameters via examples. We will provide comments on the proof of Vogan's conjecture in this setting and discuss the scope of further generalization.