Webb's conjecture and generalised Harish‐Chandra theory

Abstract Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime 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 (‐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 . Finally, we show that the condition (‐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 ‐cuspidal pairs up to conjugation.