To the memory of A. RamanathanLet G be a semisimple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Let U be the unipotent part of a Borel subgroup B ⊂ G and u the Lie algebra of U . Springer [15] has shown for good primes, that there is a B-equivariant isomorphism U → u, where B acts through conjugation on U and through the adjoint action on u (for G = SL n one has the well known equivariant isomorphism X → X − I between unipotent and nilpotent upper triangular matrices). Fix a good prime p. Then there is an isomorphism of homogeneous bundleswhere the latter can be identified with the cotangent bundle T * (G/B) of G/B. Motivated in part by [11] we establish a link between the G-invariant form χ on the Steinberg module St = H 0 (G/B, (p − 1)ρ) and Frobenius splittings [14] of the cotangent bundle T * (G/B): The representation H 0 (G/B, 2(p − 1)ρ) is a quotient of the functions H 0 (X, O X ) on X (here H 0 (G/B, M ) denotes the G-module induced from the B-module M and ρ half the sum of the roots R + opposite to the roots of B). There is a natural map ϕ : St ⊗ St → H 0 (X, O X ) such that the multiplication µ : St ⊗ St → H 0 (G/B, 2(p − 1)ρ) factors through the projection H 0 (X, O X ) → H 0 (G/B, 2(p − 1)ρ). Surprisingly the simple situation of [11] generalizes in that ϕ(a ⊗ b) is a Frobenius splitting of X if and only if χ(a ⊗ b) = 1 (if and only if µ(a ⊗ b) is a Frobenius splitting of G/B).Frobenius splitting of the cotangent bundle in this setup has a number of nice consequences. By filtering differential forms via a morphism to a suitable partial flag variety and using diagonality of the Hodge cohomology and Koszul resolutions, we obtain the vanishing theoremwhere λ is a dominant weight and Su * denotes the symmetric algebra of u * . This was proved in [1] for large dominant weights and for all dominant weights for groups of classical type and G 2 (and large primes). The simple key lemma in the very simple proof of the Borel-Bott-Weil theorem [6] implies that the vanishing theorem can be extended to weights {λ | λ, α ∨ ≥ −1, ∀α ∈ R + }. This vanishing theorem was proved in characteristic zero by Broer [3] using complete reducibility and the Borel-Bott-Weil theorem. As in characteristic zero ([3], Theorem 4.4) it follows that the subregular nilpotent variety is normal, Gorenstein and has rational singularities. In the parabolic case we prove the above vanishing theorem for regular

We characterize the globally generated, ample and very ample line bundles on Bott-Samelson varieties. Using Frobenius splitting we prove a vanishing theorem generalizing the vanishing theorem of Kumar in characteristic zero (Invent. Math. 89 (1987), 395-423).Licensed to Univ of British Columbia. Prepared on Thu Jul 2 06:00:25 EDT 2015 for download from IP 142.103.160.110.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf

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For a semisimple adjoint algebraic group G and a Borel subgroup B, consider the double classes BwB in G and their closures in the canonical compactification of G: we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically van der Kallen's filtration of the algebra of regular functions on B.