Partial character tables for ${\mathbb{Z}}_{\ell }$-spetses

Abstract

Let $\mathbb{G}$ be a simply connected ${\mathbb{Z}}_{\ell }$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of $\mathbb{G}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$. Secondly, when $\ell >2$ is a very good prime for $\mathbb{G}$, the Weyl group $W$ of $\mathbb{G}$ has order coprime with $\ell$, and $q\equiv 1(\mathrm{mod}\ell )$ we introduce a formula for the values of characters in the principal block of $\mathbb{G}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality. In both cases we formulate and provide evidence for several conjectures concerning the proposed values.