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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley Classics Library, vol. 48. Wiley, New York (1993)
Carré, C., Leclerc, B.: Splitting the square of a Schur function into its symmetric and antisymmetric parts. J. Algebraic Combin. 4, 201–231 (1995)
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)
Deligne, P., Lusztig, G.: Representations of reductive groups over a finite field. Ann. Math. 103, 103–161 (1976)
Green, J.A.: The characters of the finite general linear groups. Trans. Am. Math. Soc. 2, 402–447 (1955)
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)
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)
James, G.D., Kerber, A.: The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, vol. 16. Addison-Wesley, Reading (1981)
Kazhdan, D.: Proof of Springer’s hypothesis. Israel J. Math. 28(4), 272–286 (1977)
Kim, D.: Euler characteristic of Springer fibers. Transform. Groups. https://doi.org/10.1007/s00031-018-9487-4 (2018)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: Green polynomials and Hall–Littlewood functions at roots of unity. Eur. J. Combin. 15, 173–180 (1994)
Lusztig, G.: A class of irreducible representations of a Weyl group. Indagationes Mathematicae (Proceedings) 82, 323–335 (1979)
Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)
Lusztig, G.: Character sheaves II. Adv. Math. 57, 226–265 (1985)
Lusztig, G.: Character sheaves, V. Adv. Math. 61, 103–155 (1986)
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)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1995)
Shoji, T.: On the Springer representations of the Weyl groups of classical algebraic groups. Commun. Algebra 7(16), 1713–1745 (1979)
Shoji, T.: On the Green polynomials of classical groups. Invent. Math 74, 239–267 (1983)
Shoji, T.: Geometry of orbits and Springer correspondence. Astérisque 168, 61–140 (1988)
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)
Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)
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)
Staal, R.A.: Star diagrams and the symmetric group. Can. J. Math. 2, 79–92 (1950)
Stanley, R.P.: Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 62, vol. 2. Cambridge University Press, Cambridge (1986)
van Leeuwen, M.A.A.: A Robinson-Schensted Algorithm in the Geometry of Flags for Classical Groups. Ph.D. thesis, Rijksuniversiteit te Utrecht (1989)
van Leeuwen, M.A.A.: Some bijective correspondences involving domino tableaux. Electron. J. Combin. 7(1), 35 (2000)
Zelevinsky, A.V.: Representations of Finite Classical Groups: A Hopf Algebra Approach, Lecture Notes in Mathematics, vol. 869. Springer, Berlin (1981)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, D. On total Springer representations for classical types. Sel. Math. New Ser. 24, 4141–4196 (2018). https://doi.org/10.1007/s00029-018-0438-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-018-0438-7