The ($\hat{I}“$-asymptotic) wavefront sets: $G{L}_n$

Abstract

Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $\hat{I}“$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $\hat{I}“$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=G{L}_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $G{L}_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets.

14 pages