Global $F$-regularity of Schubert varieties with applications to $\mathcal{D}$-modules

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 ${\mathcal{D}}$-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple ${\mathcal{D}}$-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.