Multiplicities in GGGRs for Classical Type Groups with Connected Centre I

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 $G = \SO_{2n+1}(\mathbb{K})$.