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A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F-regular if the section ring S(C) = 0n>o H?(X, Cn) of an ample line bundle C on X is strongly F-regular in the sense of H?chster and Huneke [9] (cf. Definition 1.1). This important notion was introduced by Karen Smith in [18]. In this paper we prove that Schubert varieties are globally F-regular. An im mediate consequence is that local rings of Schubert varieties are strongly F-regular and thereby F-rational. Another consequence is that local rings of varieties (like de terminantal varieties) that can be identified with open subsets of Schubert varieties (cf. [13]) are strongly F-regular Let X denote a flag variety and Y C X a Schubert variety over k. Then the local cohomology sheaves HJY(Ox) are equivariant (for the action of the Borel subgroup) and holonomic (in the sense of [4]) X>x-niodules. As an application of ir rationality of Schubert varieties we apply recent results of Blickle (cf. [2]) to prove that the simple objects in the category of equivariant and holonomic Px-modules are precisely the local cohomology sheaves Hy{Ox), where c is the codimension of Y in X. Using a local Grothendieck-Cousin complex from [11], we prove that the decomposition of the local cohomology modules with support in Bruhat cells is multiplicity free (see ?4.2). In characteristic zero the local cohomology modules with support in Bruhat cells correspond to dual Verma modules. In this setting the decomposition behavior and the simple T>x -modules arise from intersection cohomology complexes of Schubert varieties by the Riemann-Hilbert correspondence. If we pick the singular codi mension one Schubert variety Y in the full flag variety Z for SL4 in characteristic zero, computations in Kazhdan-Lusztig theory show that Hy(Oz) is not a simple ?>z-module (see ?4.1).

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