Parabolic induction and the Harish-Chandra D-module

TL;DR

This paper provides explicit formulas for parabolic induction and restriction in the setting of D-modules on reductive groups, demonstrating their exactness via the flatness of the Harish-Chandra D-module over the torus.

Contribution

It introduces explicit formulas for these functors in the case of a maximal torus and proves their exactness through the flatness of the Harish-Chandra D-module.

Findings

Explicit formulas for parabolic induction and restriction.

Proof of exactness of these functors.

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