Wavefront sets and descent method for finite unitary groups

Abstract

Let $G$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$. In the 1980s, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group $G^F$ in the case where $q$ is a power of a good prime for $G^F$. An essential feature of GGGRs is that they are very closely related to the (Kawanaka) wavefront sets of the irreducible representations $\pi$ of $G^F$. In \cite[Theorem 11.2]{L7}, Lusztig showed that if a nilpotent element $X\in G^F$ is ``large'' for an irreducible representation $\pi$, then the representation $\pi$ appears with ``small'' multiplicity in the GGGR associated to $X$. In this paper, we prove that for unitary groups, if $X$ is the wavefront of $\pi$, the multiplicity equals one, which generalizes the multiplicity one result of usual Gelfand-Graev representations. Moreover, we give an algorithm to decompose GGGRs for $\textrm{U}_n(\mathbb{F}_q)$ and calculate the $\textrm{U}_4(\mathbb{F}_q)$ case by this algorithm.