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Selecta Mathematica

, Volume 24, Issue 5, pp 4141–4196 | Cite as

On total Springer representations for classical types

  • Dongkwan Kim
Article
  • 15 Downloads

Abstract

We give explicit formulas on total Springer representations for classical types. We also describe the characters of restrictions of such representations to a maximal parabolic subgroup isomorphic to a symmetric group. As a result, we give closed formulas for the Euler characteristic of Springer fibers.

Keywords

Springer fiber Springer representation Weyl group Green polynomial Kostka-Foulkes polynomial 

Mathematics Subject Classification

20C99 

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Notes

Acknowledgements

The author wishes to thank George Lusztig for having stimulating discussions with him and checking the draft of this paper. He is grateful to Jim Humphreys for his detailed remarks which help improve the readability of this paper. Also he thanks Gus Lonergan, Toshiaki Shoji, and an anonymous referee for useful comments.

References

  1. 1.
    Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley Classics Library, vol. 48. Wiley, New York (1993)Google Scholar
  2. 2.
    Carré, C., Leclerc, B.: Splitting the square of a Schur function into its symmetric and antisymmetric parts. J. Algebraic Combin. 4, 201–231 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    de Concini, C., Lusztig, G., Procesi, C.: Homology of the zero-set of a nilpotent vector field on a flag manifold. J. Am. Math. Soc. 1(1), 15–34 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Deligne, P., Lusztig, G.: Representations of reductive groups over a finite field. Ann. Math. 103, 103–161 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Green, J.A.: The characters of the finite general linear groups. Trans. Am. Math. Soc. 2, 402–447 (1955)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hotta, R., Springer, T.A.: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups. Invent. Math. 41(2), 113–127 (1977)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hotta, R., Shimomura, N.: The fixed point subvarieties of unipotent transformations on generalized flag varieties and the Green functions: combinatorial and cohomological treatments centering \({G}{L}_n\). Math. Ann. 241, 193–208 (1979)MathSciNetCrossRefGoogle Scholar
  8. 8.
    James, G.D., Kerber, A.: The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, vol. 16. Addison-Wesley, Reading (1981)Google Scholar
  9. 9.
    Kazhdan, D.: Proof of Springer’s hypothesis. Israel J. Math. 28(4), 272–286 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kim, D.: Euler characteristic of Springer fibers. Transform. Groups.  https://doi.org/10.1007/s00031-018-9487-4 (2018)
  11. 11.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Green polynomials and Hall–Littlewood functions at roots of unity. Eur. J. Combin. 15, 173–180 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lusztig, G.: A class of irreducible representations of a Weyl group. Indagationes Mathematicae (Proceedings) 82, 323–335 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lusztig, G.: Character sheaves II. Adv. Math. 57, 226–265 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lusztig, G.: Character sheaves, V. Adv. Math. 61, 103–155 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lusztig, G.: An induction theorem for Springer’s representations, representation theory of algebraic groups and quantum groups. Adv. Stud. Pure Math. 40, 253–259 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1995)Google Scholar
  18. 18.
    Shoji, T.: On the Springer representations of the Weyl groups of classical algebraic groups. Commun. Algebra 7(16), 1713–1745 (1979)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shoji, T.: On the Green polynomials of classical groups. Invent. Math 74, 239–267 (1983)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shoji, T.: Geometry of orbits and Springer correspondence. Astérisque 168, 61–140 (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Springer, T.A.: Generalization of Green’s polynomials, representation theory of finite groups and related topics. Proceedings of Symposia in Pure Mathematics, vol. XXI. American Mathematical Society, pp. 149–153 (1971)Google Scholar
  22. 22.
    Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Springer, T.A., Steinberg, R.: Conjugacy classes, Seminar on algebraic groups and related finite groups: held at the Institute for Advanced Study, Princeton\(/\)NJ, 1968\(/\)69, Lecture Notes in Mathematics, vol. 131. Springer, Berlin pp. 167–266 (1970)Google Scholar
  24. 24.
    Staal, R.A.: Star diagrams and the symmetric group. Can. J. Math. 2, 79–92 (1950)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 62, vol. 2. Cambridge University Press, Cambridge (1986)CrossRefGoogle Scholar
  26. 26.
    van Leeuwen, M.A.A.: A Robinson-Schensted Algorithm in the Geometry of Flags for Classical Groups. Ph.D. thesis, Rijksuniversiteit te Utrecht (1989)Google Scholar
  27. 27.
    van Leeuwen, M.A.A.: Some bijective correspondences involving domino tableaux. Electron. J. Combin. 7(1), 35 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zelevinsky, A.V.: Representations of Finite Classical Groups: A Hopf Algebra Approach, Lecture Notes in Mathematics, vol. 869. Springer, Berlin (1981)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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