2024/08/12 by Damiano Rossi, Rossi, Damiano
Mathematics · Physics and Astronomy · #20C20 #20G40 #55P10 #55P91 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2408.06129
openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime ℓ is contractible. This conjecture was proved by Symonds in 1998. In this paper, we suggest a generalisation of Webb's conjecture for finite reductive groups. This is done by associating to each irreducible character a new simplicial complex defined in terms of Deligne--Lusztig theory. We then show that our conjecture follows from a condition, called (e-HC-conj) below, related to generalised Harish-Chandra theory. In particular, using earlier results of the author, we prove our conjecture and recover Symonds result for finite reductive groups under mild restrictions on the prime ℓ. Finally, we show that the condition (e-HC-conj) is implied by the contractibility of the orbit spaces associated to our newly defined complex offering an unexplored topological approach to proving the uniqueness of e-cuspidal pairs up to conjugation.