Partial character tables for $\mathbb{Z}_\ell$-spetses
Radha Kessar, Gunter Malle, Jason Semeraro

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 be a simply connected -spets, let be a prime power, prime to and let be the underlying Sylow -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 on the elements of . Using this, we explicitly list the unipotent character values of the -spets related to the Benson-Solomon fusion system Sol. Secondly, when is a very good prime for , the Weyl group of has order coprime with , and we introduce a formula for the values of characters in the principal block of which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a…
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