Wavefront sets of unipotent representations of reductive 𝑝-adic groups II

Abstract The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra–Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig’s category of unipotent representations in terms of the Deligne–Langlands–Lusztig correspondence. Our formula generalises the main result of [D. Ciubotaru, L. Mason-Brown and E. Okada, Wavefront sets of unipotent representations of reductive 𝑝-adic groups I, preprint (2021), https://arxiv.org/abs/2112.14354 ], where this formula was obtained in the Iwahori-spherical case. We deduce that, for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton. In the process, we establish new properties of the generalised Springer correspondence in relation to Lusztig’s families of unipotent representations of finite reductive groups.