Webb's conjecture and generalised Harish-Chandra theory

Autor:	Rossi, Damiano
Rok vydání:	2024
Předmět:	Mathematics - Representation Theory Mathematics - Algebraic Topology Mathematics - Group Theory 20G40 20C20 55P10 55P91
Druh dokumentu:	Working Paper
Popis:	Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime $\ell$ 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 $\ell$. 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. Comment: 11 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2408.06129 Zobrazit plný text záznamu View this record from Arxiv