Abstract
For Weyl groups of classical types, we present formulas to calculate the restriction of Springer representations to a maximal parabolic subgroup of the same type. As a result, we give recursive formulas for Euler characteristics of Springer fibers for classical types. We also give tables of those for exceptional types.
Similar content being viewed by others
References
D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993.
C. de Concini, G. Lusztig, C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15–34.
M. Ehrig, C. Stroppel, 2-row Springer fibres and Khovanov diagram algebras for type D, Canad. J. Math. 68 (2016), no. 6, 1285––1333.
L. Fresse, Betti numbers of Springer fibers in type A, J. Algebra 322 (2009), 2566 – 2579.
L. Fresse, A unified approach on Springer fibers in the hook, two-row and two-column cases, Transform. Groups 15 (2010), 285 – 331.
F. Y. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan–Lusztig theory, Adv. Math. 178 (2003), 244 – 276.
R. Hotta, T. A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), no. 2, 113–127.
R. Hotta, N. Shimomura, The fixed point subvarieties of unipotent transformations on generalized ag varieties and the Green functions: Combinatorial and cohomological treatments centering GL n, Math. Ann. 241 (1979), 193–208.
M. Khovanov, Crossingless matchings and the cohomology of (n, n) Springer varieties, Commun. Contemp. Math. 6 (2004), 561–577.
F. Lübeck, Tables of Green functions for exceptional groups, available at http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/Green/index.html.
G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. Math. 42 (1981), 169–178.
G. Lusztig, Character sheaves, V, Adv. Math. 61 (1986), 103–155.
G. Lusztig, An induction theorem for Springer’s representations, in: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., Vol. 40, 2004, pp. 253–259.
J. S. Milne, Étale Cohomology, Princeton Mathematical Series, Princeton University Press, Princeton, 1980.
H. M. Russell, A topological construction for all two-row Springer varieties, Pacific J. Math. 253 (2011), no. 1, 221 – 255.
N. Shimomura, A theorem on the fixed point set of a unipotent transformation on the ag manifold, J. Math. Soc. Japan 32 (1980), no. 1, 55–64.
T. Shoji, On the Springer representations of the Weyl groups of classical algebraic groups, Comm. Algebra 7 (1979), no. 16, 1713–1745.
T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), 239–267.
N. Spaltenstein, Classes Unipotentes et Sous-groupes de Borel, Lecture Notes in Mathematics, Vol. 946, Springer-Verlag Berlin, 1982.
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207.
B. Srinivasan, Green polynomials of finite classical groups, Comm. Algebra 5 (1977), no. 12, 1241–1258.
R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1986.
C. Stroppel, B. Webster, 2-block Springer fibers: convolution algebras and coherent sheaves, Comment. Math. Helv. 87 (2012), no. 2, 477–520.
M. A. A. van Leeuwen, A Robinson–Schensted algorithm in the geometry of flags for classical groups, Ph.D. thesis, Rijksuniversiteit te Utrecht, 1989, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/pdf/thesis.pdf.
A. Wilbert, Topology of two-row Springer fibers for the even orthogonal and symplectic group, Trans. Amer. Math. Soc. 370 (2018), 2707–2737.
A. Wilbert, Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory, arXiv:1611.09828 (2016), https://doi.org/10.1007/s00209-018-2161-7
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
KIM, D. EULER CHARACTERISTIC OF SPRINGER FIBERS. Transformation Groups 24, 403–428 (2019). https://doi.org/10.1007/s00031-018-9487-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-018-9487-4