Cellularity of endomorphism algebras of tilting objects

Abstract

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer [4]. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases—and thus a general theory of “cell modules”—for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category O of the rational Cherednik algebra. For real reflection groups these bases are cellular.