Multiplicities in GGGRs for Classical Type Groups with Connected Centre I

TL;DR

This paper advances the explicit computation of multiplicities involving generalized Gelfand–Graev representations for classical type groups with connected centers, focusing on the relation between character sheaves and almost characters under certain conditions.

Contribution

It completes the first step in computing multiplicities by explicitly relating characteristic functions of character sheaves to almost characters for classical groups with connected centers.

Findings

Computed scalars relating characteristic functions of character sheaves to almost characters

Extended Lusztig's method from special orthogonal groups to broader classical groups

Provided explicit formulas under restrictions on the prime power q

Abstract

Assume $G$ is a connected reductive algebraic group defined over $\bar{\mathbb{F}_p}$ such that $p$ is good prime for $G$. Furthermore we assume that $Z(G)$ is connected and $G/Z(G)$ is simple of classical type. Let $F$ be a Frobenius endomorphism of $G$ admitting an $\mathbb{F}_q$-rational structure $G^F$. This paper is one of a series whose overall goal is to compute explicitly the multiplicity $< D_0745664 {G^F}(\Gamma_u),\chi>$ where: $\chi$ is an irreducible character of $G^F$, $D_{G^F}(\Gamma_u)$ is the Alvis--Curtis dual of a generalised Gelfand--Graev representation of $G^F$ and $u \in G^F$ is contained in the unipotent support of $\chi$. In this paper we complete the first step towards this goal. Namely we explicitly compute, under some restrictions on $q$, the scalars relating the characteristic functions of character sheaves of $G$ to the almost characters of $G^F$ whenever the support of the character sheaf contains a unipotent element. We achieve this by adapting a method of Lusztig who answered this question when $G$ is a special orthogonal group $\SO_{2n+1}(\mathbb{K})$. Consequently the main result of this paper is due to Lusztig when $G = \SO_{2n+1}(\mathbb{K})$.