Abstract
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.
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References
Andersen, H.H.: The first cohomology group of line bundle on G/B. Invent. Math. 51, 287–296 (1979)
Anwar, M.F.: On the cohomology of certain homogeneous vector bundles of G/B in characteristic zero. J. Pure Appl. Algebra 216(5), 1160–1163 (2012)
Anwar, M.F.: Representations and cohomology of algebraic groups. PhD thesis, University of York, UK (2011)
Demazure, M.: A very simple proof of bott’s theorem. Invent. math. 33, 271–272 (1976)
Donkin, S.: The cohomology of line bundles on the three-dimensional flag variety. J. Algebra 307, 570–613 (2007)
Donkin, S.: Rational representations of algebraic groups: tensor products and filtrations. Lecture Notes in Math, vol. 1140. Springer, Berlin (1985)
Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. Journal fü,r die reine und angewandte Mathematik 317, 157–199 (1980)
Jantzen, J.C.: Representations of algebraic groups. In: Math. Surveys Monogr., vol. 107. Amer. Math. Society. 2nd edn. (2003)
Acknowledgments
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.
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Presented by: Peter Littelmann
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Anwar, M.F. On the Cohomology of Certain Rank 2 Vector Bundles on G/B. Algebr Represent Theor 22, 1101–1108 (2019). https://doi.org/10.1007/s10468-018-9813-5
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DOI: https://doi.org/10.1007/s10468-018-9813-5