A Microlocal Description of Aubert-Zelevinsky Duality on Unipotent $L$-Parameters
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 on an endoscopic subgroup: Fourier transform, Chevalley involution and duality on local systems. When the group is not inner to an unramified triality form of $D_4$ we further show that one does not need to pass to an endoscopic subgroup. 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 those examples since only self-dual local systems appear. Motivated by categorical considerations, we provide a second formulation, involving complex conjugation from the compact form of the dual group, giving a covariant functor of perverse sheaves that agrees with Aubert--Zelevinsky duality on $L$-parameters and as involutions of graded Hecke algebras. This formulation holds without passing to an endoscopic subgroup and is valid for all inner forms of simple adjoint unramified groups. Finally, we prove the microlocal Hiraga conjecture for unipotent $A$-parameters of inner-to-split simple adjoint groups as a consequence of our results.