Partial character tables for ℤ_ℓ-spetses

2025/07/11 by Kessar, Radha, Malle, Gunter, Semeraro, Jason
#16G30 #20C08 #20C20 #20D20 #20F55 #55R35 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.08502

Abstract

Let \mathbbG be a simply connected ℤ_ℓ-spets, let q be a prime power, prime to ℓ and let S be the underlying Sylow ℓ-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 \mathbbG(q) on the elements of S. Using this, we explicitly list the unipotent character values of the ℤ₂-spets G₂₄(q) related to the Benson-Solomon fusion system Sol(q). Secondly, when ℓ > 2 is a very good prime for \mathbbG, the Weyl group W of \mathbbG has order coprime with ℓ, and q≡1\pmodℓ we introduce a formula for the values of characters in the principal block of \mathbbG(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.

Citations

• Weight conjectures for ℓ-compact groups and spetses
• A 2-compact group as a spets
• The development version of the CHEVIE package of GAP3
• Split Spetses for primitive reflection groups

Related

• A 2-compact group as a spets
• The values of unipotent characters at unipotent elements for groups of type E₈ and ² E₆
• Weight conjectures for ℓ-compact groups and spetses
• Constructing characters of Sylow p-subgroups of finite Chevalley groups
• On e-local structures for ℤ_ℓ-spetses