Parabolic induction and the Harish-Chandra ????-module

Let G G be a reductive group and L L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad \operatorname {Ad} -equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D {\mathscr {D}} -modules on G G and L L , respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L = T L=T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D {\mathscr {D}} -module on G × T {G\times T} . We show that this module is flat over D ( T ) {\mathscr {D}}(T) , which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of D {\mathscr {D}} -modules.