On the Coxeter complex and Alvis-Curtis duality for principal $\ell$-blocks
of $GL_n(q)$
Markus Linckelmann and Sibylle Schroll

Abstract. M. Cabanes and J. Rickard showed in [2] that the Alvis-Curtis
character duality of a finite group of Lie type is induced in non defining
characteristic $\ell$ 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 the principal blocks of
general linear groups with abelian Sylow-$\ell$-subgroups that this is true,
by an explicit verification relating the complex $X$ to the Coxeeter complex
of the corresponding Weyl group.