On Unipotent Supports of Reductive Groups With a Disconnected Centre
Résumé
Let $\mathbf{G}$ be a connected reductive algebraic group defined over an algebraic closure of the finite field of prime order $p>0$, which we assume to be good for $\mathbf{G}$. We denote by $F : \mathbf{G} \to \mathbf{G}$ a Frobenius endomorphism of $\mathbf{G}$ and by $G$ the corresponding $\mathbb{F}_q$-rational structure. If $\operatorname{Irr}(G)$ denotes the set of ordinary irreducible characters of $G$ then by work of Lusztig and Geck we have a well defined map $\Phi_{\mathbf{G}} : \operatorname{Irr}(G) \to \{F\text{-stable unipotent conjugacy classes of }\mathbf{G}\}$ where $\Phi_{\mathbf{G}}(\chi)$ is the unipotent support of $\chi$.
Lusztig has given a classification of the irreducible characters of $G$ and obtained their degrees. In particular he has shown that for each $\chi \in \operatorname{Irr}(G)$ there exists an integer $n_{\chi}$ such that $n_{\chi}\cdot\chi(1)$ is a monic polynomial in $q$. Given a unipotent class $\mathcal{O}$ of $\mathbf{G}$ with representative $u \in \mathbf{G}$ we may define $A_{\mathbf{G}}(u)$ to be the finite quotient group $C_{\mathbf{G}}(u)/C_{\mathbf{G}}(u)^{\circ}$. If the centre $Z(\mathbf{G})$ is connected and $\mathbf{G}/Z(\mathbf{G})$ is simple then Lusztig and H\'{e}zard have independently shown that for each $F$-stable unipotent class $\mathcal{O}$ of $\mathbf{G}$ there exists $\chi \in \operatorname{Irr}(G)$ such that $\Phi_{\mathbf{G}}(\chi)=\mathcal{O}$ and $n_{\chi} = |A_{\mathbf{G}}(u)|$, (in particular the map $\Phi_{\mathbf{G}}$ is surjective).
The main result of this thesis extends this result to the case where $\mathbf{G}$ is any simple algebraic group, (hence removing the assumption that $Z(\mathbf{G})$ is connected). In particular if $\mathbf{G}$ is simple we show that for each $F$-stable unipotent class $\mathcal{O}$ of $\mathbf{G}$ there exists $\chi \in \operatorname{Irr}(G)$ such that $\Phi_{\mathbf{G}}(\chi) = \mathcal{O}$ and $n_{\chi} = |A_{\mathbf{G}}(u)^F|$ where $u \in \mathcal{O}^F$ is a well-chosen representative. We then apply this result to prove, (for most simple groups), a conjecture of Kawanaka's on generalised Gelfand--Graev representations (GGGRs). Namely that the GGGRs of $G$ form a $\mathbf{Z}$-basis for the $\mathbf{Z}$-module of all unipotently supported class functions of $G$. Finally we obtain an expression for a certain fourth root of unity associated to GGGRs in the case where $\mathbf{G}$ is a symplectic or special orthogonal group.