On parabolic restriction of perverse sheaves

TL;DR

This paper proves the exactness of certain functors related to perverse sheaves on reductive groups, extends known results, and proposes a conjecture with supporting evidence, broadening the understanding of sheaf theory in algebraic groups.

Contribution

It generalizes Lusztig's exactness result for character sheaves to conjugation equivariant sheaves and introduces a conjecture on Harish-Chandra transform's t-exactness with supporting evidence.

Findings

Proved exactness of parabolic restriction and induction functors.

Proposed and provided evidence for a conjecture on Harish-Chandra transform.

Extended results on perverse sheaves from tori to reductive groups.

Abstract

We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.