Parabolic induction and the Harish-Chandra $\boldsymbol {\mathcal {D}}$-module

Abstract:

Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $\operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${\mathscr {D}}$-modules on $G$ and $L$, 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 $L=T$ is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra ${\mathscr {D}}$-module on ${G\times T}$. We show that this module is flat over ${\mathscr {D}}(T)$, which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of ${\mathscr {D}}$-modules.