Algebras and Representation Theory

, Volume 22, Issue 5, pp 1101–1108 | Cite as

On the Cohomology of Certain Rank 2 Vector Bundles on G/B

  • M. Fazeel AnwarEmail author


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.


Algebraic 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.


  1. 1.
    Andersen, H.H.: The first cohomology group of line bundle on G/B. Invent. Math. 51, 287–296 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anwar, M.F.: Representations and cohomology of algebraic groups. PhD thesis, University of York, UK (2011)Google Scholar
  4. 4.
    Demazure, M.: A very simple proof of bott’s theorem. Invent. math. 33, 271–272 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Donkin, S.: The cohomology of line bundles on the three-dimensional flag variety. J. Algebra 307, 570–613 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Donkin, S.: Rational representations of algebraic groups: tensor products and filtrations. Lecture Notes in Math, vol. 1140. Springer, Berlin (1985)CrossRefGoogle Scholar
  7. 7.
    Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. Journal fü,r die reine und angewandte Mathematik 317, 157–199 (1980)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Jantzen, J.C.: Representations of algebraic groups. In: Math. Surveys Monogr., vol. 107. Amer. Math. Society. 2nd edn. (2003)Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSukkur IBA UniversitySukkurPakistan

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