Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations

TL;DR

This paper investigates the structure and properties of ABV-packets for p-adic groups, demonstrating their independence from Whittaker data, establishing their relation to L-packets and Arthur packets, and proving Vogan's conjecture for tempered representations.

Contribution

It proves the independence of ABV-packets from Whittaker data, establishes their equivalence with L-packets and Arthur packets for tempered parameters, and confirms Vogan's conjecture in this context.

Findings

ABV-packets do not depend on Whittaker datum

ABV-packets for tempered parameters are Arthur packets

Normalized vanishing cycles coincide with Langlands and Arthur parameters

Abstract

We show that ABV-packets for $p$-adic groups do not depend on the choice of a Whittaker datum, but the function from the ABV-packet to representations of the appropriate microlocal equivariant fundamental group does, and we find this dependence exactly. We study the relation between open parameters and tempered parameters and Arthur parameters and generic representations. We state a genericity conjecture for ABV-packets and prove this conjecture for quasi-split classical groups and their pure inner forms. Motivated by this we study ABV-packets for open parameters and prove that they are L-packets, and further that the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by the Langlands correspondence. From this conclude Vogan's conjecture on A-packets for tempered representations: ABV-packets for tempered parameters are Arthur packets and the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by Arthur.