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Inductive McKay condition for finite simple groups of type $ \mathsf{C}$

Authors: Marc Cabanes and Britta Späth
Journal: Represent. Theory 21 (2017), 61-81
MSC (2010): Primary 20C15, 20C33; Secondary 20G40
Published electronically: June 14, 2017
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Abstract: We verify the inductive McKay condition for simple groups of Lie type $ \mathsf {C}$, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow $ d$-tori in those groups.

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  • [BM92] Michel Broué and Gunter Malle, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis, Math. Ann. 292 (1992), no. 2, 241-262 (French). MR 1149033,
  • [BMi97] Michel Broué and Jean Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 73-139 (French). MR 1429870
  • [CE] Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756
  • [CS13] Marc Cabanes and Britta Späth, Equivariance and extendibility in finite reductive groups with connected center, Math. Z. 275 (2013), no. 3-4, 689-713. MR 3127033,
  • [CS17] M. Cabanes and B. Späth, Equivariant character correspondences and inductive McKay condition for type A, arXiv:1305.6407. To appear in J. Reine Angew. Math., 2017.
  • [FS89] Paul Fong and Bhama Srinivasan, The blocks of finite classical groups, J. Reine Angew. Math. 396 (1989), 122-191. MR 988550
  • [GLS] Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592
  • [I] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR 0460423
  • [IMN07] I. M. Isaacs, Gunter Malle, and Gabriel Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), no. 1, 33-101. MR 2336079,
  • [LS] Martin W. Liebeck and Gary M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, vol. 180, American Mathematical Society, Providence, RI, 2012. MR 2883501
  • [L77] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125-175. MR 0463275,
  • [L88] G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157-166. Orbites unipotentes et représentations, I. MR 1021495
  • [M07] Gunter Malle, Height 0 characters of finite groups of Lie type, Represent. Theory 11 (2007), 192-220. MR 2365640,
  • [MS16] Gunter Malle and Britta Späth, Characters of odd degree, Ann. of Math. (2) 184 (2016), no. 3, 869-908. MR 3549625,
  • [Mi69] John Milnor, On isometries of inner product spaces, Invent. Math. 8 (1969), 83-97. MR 0249519,
  • [S09] Britta Späth, The McKay conjecture for exceptional groups and odd primes, Math. Z. 261 (2009), no. 3, 571-595. MR 2471089,
  • [S10a] Britta Späth, Sylow $ d$-tori of classical groups and the McKay conjecture. I, J. Algebra 323 (2010), no. 9, 2469-2493. MR 2602390,
  • [S10b] Britta Späth, Sylow $ d$-tori of classical groups and the McKay conjecture. II, J. Algebra 323 (2010), no. 9, 2494-2509. MR 2602391,
  • [S12] Britta Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426-438. MR 2966987,
  • [Sp74] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159-198. MR 0354894,
  • [T16] J. Taylor,
    Action of automorphisms on irreducible characters of symplectic groups, arXiv:1612.03138, 2016.
  • [W63] G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1-62. MR 0150210

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Additional Information

Marc Cabanes
Affiliation: CNRS, IMJ-PRG, Boite 7012, 75205 Paris Cedex 13, France

Britta Späth
Affiliation: Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Keywords: McKay's conjecture, symplectic groups, action of automorphisms on characters, extended Weyl groups
Received by editor(s): September 16, 2016
Received by editor(s) in revised form: March 29, 2017
Published electronically: June 14, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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