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

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.

Equivariant Functors and Sheaves

“…The equivariant derived category is an important object in arithmetic geometry and representation theory, as it provides the natural location in which the theories of equivariant (ℓ-adic) sheaves, equivariant (ℓ-adic) local systems, and equivariant (ℓ-adic) perverse sheaves (cf. [66], [63], [2], [28], [46], [67], [8], [65]). However, while the notion of equivariance is straightforward to define and follow the definition as given by Mumford in [77] (namely that an object F in a category C (X) of sheaves of some sort on X is equivariant if, for an action morphism α X : G × X → X and projection π 2 : G × X → X for an algebraic group G and a left G-variety X, there is an isomorphism θ from the pullback of F along α X and the pullback of F along the projection map in C (G × X)), but the notion of equivariance is much more subtle and difficult to extend to the full derived category level.…”

“…[102], [1], [19]), the equivariant derived category introduced in [65], D b G (X), has been used in various places. It was developed to use a type of equivariant cohomology to construct the graded Hecke algebra; this equivariant derived category and its connections to graded Hecke algebras has been used in, for example, [35], [51], and crucially in [28] to give an equivariant notion of Brylinski's Ev functor and equivariant nearby and vanishing cycles.…”

“…This extension restricts the resolutions, however, from all free G-spaces to instead acyclic G-resolutions of the scheme/variety X. This approach to the equivariant bound derived category has seen use in [28], [46], [49], [55], [75], and [87], for instance.…”

“…The equivariant derived category of Lusztig is a given as a construction which takes a G-scheme X and resolves the action of G on X by a certain class of smooth free principal G-varieties which nicely control the singularities of the action and then takes descent data over these resolutions in order to produce the equivariance that equivaraint derived complexes need to have. This approach to equivariance is the one primarily used in [28] to define equivariant vanishing and nearby cycles. The equivariant derived category of Lusztig is a formally powerful construction (and may be readily generalized to equivariant categories indexed by pseudofunctors as in Section 3 above), but it also is intimately related to equivariant cohomology and homology of [64,65] and is used in [65] to construct the graded affine Hecke algebra.…”