A Microlocal Description of Aubert-Zelevinsky Duality on Unipotent $L$-Parameters

TL;DR

This paper provides a microlocal perspective on Aubert-Zelevinsky duality for unipotent representations of p-adic groups, linking it to Fourier transforms, involutions, and local systems, and proves the microlocal Hiraga conjecture.

Contribution

It introduces a unified microlocal description of Aubert-Zelevinsky duality and clarifies the structure of unipotent local Langlands correspondence for p-adic groups.

Findings

Microlocal description of Aubert-Zelevinsky involution via Fourier transform and conjugation.

Simplification of the duality composition for non-triality forms of D4.

Proof of the microlocal Hiraga conjecture for unipotent A-parameters.

Abstract

We give a microlocal description of the Aubert--Zelevinsky involution for all unipotent representations of all inner forms of simple adjoint unramified $p$-adic groups. Via the realization of enhanced $L$-parameters as perverse sheaves, we show that the involution corresponds to the composition of three operations: Fourier transform, the complex conjugation map coming from the compact form of the dual group, and inversion on the compact part of the infinitesimal parameter. We also show that when the group is not inner to a triality form of $D_4$, this simplifies to the composition of Fourier transform, Chevalley involution, and duality on local systems. This was previously verified in certain special examples by several authors where only the contribution by Chevalley involution and Fourier transform was observed. Duality on local systems is invisible in these examples since they only involve self-dual local systems. Finally, we prove the microlocal Hiraga conjecture for unipotent $A$-parameters of inner-to-split simple adjoint groups as a consequence of our results. In order to give a uniform proof of our results we reformulate and clarify several aspects of the construction of the unipotent local Langlands correspondence. This additionally allows us to characterize how various affine and graded Hecke algebras are identified. We prove that there is a `canonical' way to do so by showing that there is a unique isomorphism of graded Hecke algebras compatible with the Kottwitz isomorphism. As an application of this, we show that a simple module of the geometric graded Hecke algebra is uniquely determined by certain composition multiplicities coming from the corresponding representation of the $p$-adic group. This can be understood as a characterization of the unipotent local Langlands correspondence.