Parabolic induction and the Harish-Chandra 𝒟-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 Ad -equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D {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 {D} -module on G × T {G× T} . We show that this module is flat over D ( T ) {D}(T) , which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of D {D} -modules.