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Register a new userArthur packets for $G_2$ and perverse sheaves on cubics
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This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.
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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 Arthurs main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogans 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.
When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ of $W$-equivariant perverse sheaves on $E_{mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtmans recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.
Let $Q$ be a finite quiver without loops and $mathcal{Q}_{alpha}$ be the Lusztig category for any dimension vector $alpha$. The purpose of this paper is to prove that all Frobenius eigenvalues of the $i$-th cohomology $mathcal{H}^i(mathcal{L})|_x$ for a simple perverse sheaf $mathcal{L}in mathcal{Q}_{alpha}$ and $xin mathbb{E}_{alpha}^{F^n}=mathbb{E}_{alpha}(mathbb{F}_{q^n})$ are equal to $(sqrt{q^n})^{i}$ as a conjecture given by Schiffmann (cite{Schiffmann2}). As an application, we prove the existence of a class of Hall polynomials.
We begin this paper by reviewing the Langlands correspondence for unipotent representations of the exceptional group of type $G_2$ over a $p$-adic field $F$ and present it in an explicit form. Then we compute all ABV-packets, as defined in [CFM+21] following ideas from Vogans 1993 paper The local Langlands Conjecture, and prove that these packets satisfy properties derived from the expectation that they are generalized A-packets. We attach distributions to ABV-packets for $G_2$ and its endoscopic groups and study a geometric endoscopic transfer of these distributions. This paper builds on earlier work by the same authors.
We establish a Springer correspondence for classical symmetric pairs making use of Fourier transform, a nearby cycle sheaf construction and parabolic induction. In particular, we give an explicit description of character sheaves for classical symmetric pairs.
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