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 $w\in W$, 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 $D^b(O_0^Z)$ of the $Z$-graded BGG category $O_0^Z$ via derived twisting functors as well as a categorical action of $H(W)$ on $D^b(O_0^Z)$ via derived shuffling functors. As applications, we get graded character formulae for $T_sL(x)$ and $C_sL(x)$ for each simple reflection $s$. We describe the graded shifts occurring in the action of the $Z$-graded twisting and shuffling functors on dual Verma modules and simple modules. We also characterize the action of the derived $Z$-graded Zuckerman functors on simple modules.