The (-asymptotic) wavefront sets: GLₙ

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 -asymptotic wavefront sets generalizing the notion of wavefront sets via the -asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When G=GLₙ, 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 GLₙ, 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.