Toward the endoscopic classification of unipotent representations of $p$-adic $G_2$

TL;DR

This paper advances the understanding of unipotent representations of the exceptional p-adic group G_2 by explicitly classifying ABV-packets, analyzing their properties, and exploring endoscopic transfer mechanisms.

Contribution

It provides an explicit classification of ABV-packets for G_2's unipotent representations and investigates their endoscopic transfer properties, extending prior theoretical frameworks.

Findings

Explicit computation of ABV-packets for G_2

Verification that ABV-packets satisfy properties of generalized A-packets

Analysis of geometric endoscopic transfer of distributions

Abstract

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 Vogan's 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.