Global F-regularity of Schubert varieties with applications to D-modules

TL;DR

This paper proves that Schubert varieties are globally F-regular, leading to new insights into the structure of D-modules on flag varieties in positive characteristic, including simplification of their decomposition.

Contribution

It establishes the global F-regularity of Schubert varieties and applies this to analyze D-modules, showing simple D-modules coincide with local cohomology sheaves and are multiplicity free.

Findings

Schubert varieties are globally F-regular.

Simple D-modules coincide with local cohomology sheaves.

Local cohomology sheaves decompose multiplicity free.

Abstract

We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.