On Okuyama's theorems about Alvis-Curtis duality

The purpose of this paper is to report on the unpublished manuscript [O] by T. Okuyama where are proved some conjectures generalizing to homotopy categories the theorems of [CaRi] and [LS] holding in derived categories. We refer to the latter references and [CaEn]~\S 4 for a broader introduction to the subject. The main theme is the one of complexes related with the Coxeter complex and the action of parabolic subgroups on them, either for finite groups with BN-pairs or for finite dimensional Hecke algebras. Okuyama's contractions prove a quite efficient tool in a number of situations (see the proof of Solomon-Tits theorem in \S~6). We often stray away from Okuyama's proofs when it allows simplifications.