On the Cohomology of Certain Rank 2 Vector Bundles on G/B
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Let G be a semisimple, simply connected, linear algebraic group over an algebraically closed field k. Donkin (In J. Algebra, 307, 570–613 2007), Donkin gave a recursive description for the characters of the cohomology of line bundles on the three dimensional flag variety in prime characteristic. The recursion involves not only line bundles but also certain natural rank 2 bundles associated to two dimensional B −modules Nα(λ), where λ in an integral weight and α is a simple root. In this paper we compute the cohomology of these rank 2 bundles and simplify the recursion in Donkin (In J. Algebra, 307, 570–613 2007). We also compute the socle of Nα(λ) and give a rank 2 version of Kempf’s vanishing theorem.
KeywordsAlgebraic groups Flag varieties Cohomology Vector bundles
Mathematics Subject Classification (2010)20G05 20G10 20G15 17B10
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I am very grateful to Stephen Donkin for his suggestions and valuable remarks. I would also like to express my gratitude to H. H, Andersen for his remarks about Proposition 3.4. The research was partially supported by EPSRC grant EP/L005328/1 held by Michael Bate to whom I am also grateful for hospitality.