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

Let G be a complex semisimple Lie algebra with Weyl group W. Let H(W) be the Iwahori–Hecke algebra associated to W. For each w E W, let T_w and C_w be the corresponding Z-graded twisting functor and Z-graded shuffling functor, respectively. In this paper, we present a categorical action of H(W) on the derived category D^b(O₀^Z) of the Z-graded BGG category O₀^Z via derived twisting functors as well as a categorical action of H(W) on D^b(O₀^Z) via derived shuffling functors. As applications, we get graded character formulae for T_sL(x) and C_sL(x) for each simple reflection s.