A two-sided q-analogue of the Coxeter complex

Linckelmann, M. & Schroll, S. (2005). A two-sided q-analogue of the Coxeter complex. Journal of Algebra, 289(1), pp. 128-134. doi: 10.1016/j.jalgebra.2005.03.026,

Abstract

M. Cabanes and J. Rickard showed in [3] that the Alvis-Curtis character duality of a finite group of Lie type is induced in non defining characteristic ℓ by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-ℓ-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group.