Rigidity of Bott–Samelson–Demazure–Hansen variety for \({\varvec{F}}_{\varvec{4}}\) and \({\varvec{G}}_{\varvec{2}}\)


Let G be a simple algebraic group of adjoint type over \({\mathbb {C}},\) whose root system is of type \(F_{4}.\) Let T be a maximal torus of G and B be a Borel subgroup of G containing T. Let w be an element of the Weyl group W and X(w) be the Schubert variety in the flag variety G/B corresponding to w. Let \(Z(w, {\underline{i}})\) be the Bott–Samelson–Demazure–Hansen variety (the desingularization of X(w)) corresponding to a reduced expression \({\underline{i}}\) of w. In this article, we study the cohomology modules of the tangent bundle on \(Z(w_{0}, {\underline{i}}),\) where \(w_{0}\) is the longest element of the Weyl group W. We describe all the reduced expressions of \(w_{0}\) in terms of a Coxeter element such that \(Z(w_{0}, {\underline{i}})\) is rigid (see Theorem 7.1). Further, if G is of type \(G_{2},\) there is no reduced expression \({\underline{i}}\) of \(w_{0}\) for which \(Z(w_{0}, {\underline{i}})\) is rigid (see Theorem 8.2).

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  1. 1.

    Balaji V, Senthamarai Kannan S and Subrahmanyam K V, Cohomology of line bundles on Schubert varieties – I, Transformation Groups 9(2) (2004) 105–131

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bott R and Samelson H, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964–1029

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brion M and Kumar S, Frobenius Splitting Methods in Geometry and Representation theory, Progress in Mathematics, vol. 231 (2005) (Boston, MA: Birkhäuser, Boston Inc.)

  4. 4.

    Demazure M, Desingularisation des varieties de Schubert generalisees, Ann. Sci. Ecole Norm. Sup. 7 (1974) 53–88

    Article  Google Scholar 

  5. 5.

    Demazure M, A very simple proof of Bott’s theorem, Invent. Math. 33 (1976) 271–272

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hansen H C, On cycles on flag manifolds, Math. Scand. 33 (1973) 269–274

    MathSciNet  Article  Google Scholar 

  7. 7.

    Humphreys J E, Introduction to Lie algebras and Representation Theory (1972) (Berlin: Springer)

  8. 8.

    Humphreys J E, Linear Algebraic Groups (1975) (Berlin: Springer)

  9. 9.

    Humphreys J E, Reflection Groups and Coxeter Groups, vol. 29 (1992) (Cambridge: Cambridge University Press)

  10. 10.

    Humphreys J E, Conjugacy classes in semisimple algebraic groups, Math. Surveys Monographs, vol. 43 (1995) (Amer. Math. Soc.)

  11. 11.

    Huybrechts D, Complex Geometry: An Introduction (2005) (Berlin: Springer)

  12. 12.

    Jantzen J C, Representations of Algebraic Groups, second edition, Mathematical Surveys and Monographs, vol. 107 (2003)

  13. 13.

    Narasimha Chary B, Senthamarai Kannan S and Parameswaran A J, Automorphism group of a Bott–Samelson–Demazure–Hansen variety, Transformation Groups 20(3) (2015) 665–698

    MathSciNet  Article  Google Scholar 

  14. 14.

    Narasimha Chary B and Senthamarai Kannan S, Rigidity of Bott–Samelson–Demazure–Hansen variety for $PSp(2n,{\mathbb{C}}),$ J. Lie Theory 27(2) (2017) 435–468

    MathSciNet  Google Scholar 

  15. 15.

    Senthamarai Kannan S, On the automorphism group of a smooth Schubert variety, Algebr. Represent. Theory 19(4) (2016) 761–782.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Senthamarai Kannan S and Saha Pinakinath, Rigidity of Bott–Samelson–Demazure–Hansen variety for $PSO(2n+1,{\mathbb{C}})$, preprint

  17. 17.

    Yang S W and Zelevinsky A, Cluster algebras of finite type via Coxeter elements and principal minors, Transformation Groups 13(3–4) (2008) 855–895

    MathSciNet  Article  Google Scholar 

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The authors would like to thank the Infosys Foundation for the partial financial support.

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Correspondence to S Senthamarai Kannan.

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Communicating Editor: Nitin Nitsure

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Kannan, S.S., Saha, P. Rigidity of Bott–Samelson–Demazure–Hansen variety for \({\varvec{F}}_{\varvec{4}}\) and \({\varvec{G}}_{\varvec{2}}\). Proc Math Sci 130, 19 (2020). https://doi.org/10.1007/s12044-019-0535-3

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  • Bott–Samelson–Demazure–Hansen variety
  • coexeter element
  • tangent bundle

2000 Mathematics Subject Classification

  • 14M15