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

TL;DR

This paper develops formulas for unipotent character values of $Z_ell$-spetses and extends known character value formulas for groups of Lie type, supported by explicit examples and conjectures.

Contribution

It introduces new formulas for unipotent character values of $Z_ell$-spetses and extends Curtis-Schewe type formulas to this setting, with evidence for several conjectures.

Findings

Explicit unipotent character values for $G_{24}(q)$ related to Benson-Solomon system.

A new formula for principal block character values when $ell > 2$ and $q mod ell=1$.

Evidence supporting conjectures on character value formulas for $Z_ell$-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 $q\equiv1\pmod\ell$ 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.