The Wavefront Sets of Iwahori-Spherical Representations of Reductive $p$-adic Groups

The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an explicit description of the wavefront set of an irreducible spherical representation with real Satake parameter.Comment: Previous version, titled "Some Unipotent Arthur Packets of Reductive p-adic Groups I", featured two main results: a computation of wavefront sets and a characterization of some unipotent Arthur packets. This version only includes the former (the discussion of Arthur packets will be the subject of a separate, forthcoming paper