On Hecke algebras and ℤ-graded twisting, shuffling and Zuckerman functors

Let [Formula: see text] be a complex semisimple Lie algebra with Weyl group [Formula: see text]. Let [Formula: see text] be the Iwahori–Hecke algebra associated to [Formula: see text]. For each [Formula: see text], let [Formula: see text] and [Formula: see text] be the corresponding [Formula: see text]-graded twisting functor and [Formula: see text]-graded shuffling functor, respectively. In this paper, we present a categorical action of [Formula: see text] on the derived category [Formula: see text] of the [Formula: see text]-graded BGG category [Formula: see text] via derived twisting functors as well as a categorical action of [Formula: see text] on [Formula: see text] via derived shuffling functors. As applications, we get graded character formulae for [Formula: see text] and [Formula: see text] for each simple reflection [Formula: see text].