Wave-front set of some representations of unipotent reduction of the group SO(2n+1)

Abstract: Let G be a connected reductive group over a p-adic field F and let $\pi$ be an irreducible admissible representation of $G(\overline{F})$. Due to Harish-Chandra, there is a development of the character of $\pi$ near the origin and we can use it to define conjecturally the wave-front set of $\pi$. If this wave-front set exists, it is an unipotent orbit in $G(\overline{F})$, where $\overline{F}$ is an algebraic closure of F. Here we consider the group $G = SO(2n + 1)$ and we assume that $\pi$ is of unipotent reduction. Lusztig has defined and parametrized these representations. We assume moreover that $\pi$ is tempered or that $\pi$ is the image of a tempered representation by the Aubert-Zelevinsky’s involution. Under these assumptions, we prove that $\pi$ has a wave-front set, which is computable using the Arthur’s parameter of $\pi$.