A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F -regular if the section ring S(L) = n≥0 H 0 (X, L n ) of an ample line bundle L on X is strongly F -regular in the sense of Hochster 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 immediate consequence is that local rings of Schubert varieties are

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... y F -regular and thereby F -rational. Another consequence is that local rings of varieties (like determinantal 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 ⊂ X a Schubert variety over k. Then the local cohomology sheaves H j Y (O X ) are equivariant (for the action of the Borel subgroup) and holonomic (in the sense of [4]) D X -modules. As an application of Frationality of Schubert varieties we apply recent results of Blickle (cf. [2]) to prove that the simple objects in the category of equivariant and holonomic D X -modules are precisely the local cohomology sheaves H c Y (O X ), 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 D X -modules arise from intersection cohomology complexes of Schubert varieties by the Riemann-Hilbert correspondence. If we pick the singular codimension one Schubert variety Y in the full flag variety Z for SL 4 in characteristic zero, computations in Kazhdan-Lusztig theory show that H 1 Y (O Z ) is not a simple D Z -module (see §4.1). Global F -regularity and Frobenius splitting Let k denote a field of characteristic p > 0 and R a finitely generated k-algebra. For an R-module M we define F e * M to be the R-module which is equal to M as an abelian group with R-action given by r · m = r p e m. Definition 1.1 ([9]). The ring R is said to be strongly F -regular if for every c ∈ R, not contained in any minimal prime of R, there exists a positive integer e ≥ 0 such