Characters of odd degree

Abstract

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime $2$, hence for characters of odd degree.

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      doi = {10.1007/s00209-008-0340-7},
      journal = {Math. Z.},
      zblnumber = {1178.20006},
      volume = {261},
      mrnumber = {2471089},
      fjournal = {Mathematische Zeitschrift},
      mrreviewer = {Stuart Martin},
      coden = {MAZEAX},
      title = {The {M}c{K}ay conjecture for exceptional groups and odd primes},
      year = {2009},
      pages = {571--595},
      }
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    @ARTICLE{Sp13, mrkey = {3100954},
      issn = {0075-4102},
      author = {Sp{ä}th, Britta},
      mrclass = {20C20 (20C15 20C33)},
      doi = {10.1515/crelle.2012.035},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1283.20006},
      volume = {680},
      mrnumber = {3100954},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      mrreviewer = {Burkhard K{ü}lshammer},
      title = {A reduction theorem for the {A}lperin-{M}c{K}ay conjecture},
      year = {2013},
      pages = {153--189},
      }
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    @ARTICLE{Spaeth2, mrkey = {2602390},
      number = {9},
      issn = {0021-8693},
      author = {Sp{ä}th, Britta},
      mrclass = {20C33 (20C20 20G05 20G40)},
      doi = {10.1016/j.jalgebra.2010.02.008},
      journal = {J. Algebra},
      zblnumber = {1260.20015},
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      mrnumber = {2602390},
      fjournal = {Journal of Algebra},
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      coden = {JALGA4},
      title = {Sylow {$d$}-tori of classical groups and the {M}c{K}ay conjecture. {I}},
      year = {2010},
      pages = {2469--2493},
      }
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    @ARTICLE{Sp12, mrkey = {2966987},
      number = {3},
      issn = {0024-6093},
      author = {Sp{ä}th, Britta},
      mrclass = {20C33 (20C15)},
      doi = {10.1112/blms/bdr100},
      journal = {Bull. Lond. Math. Soc.},
      zblnumber = {1251.20020},
      volume = {44},
      mrnumber = {2966987},
      fjournal = {Bulletin of the London Mathematical Society},
      mrreviewer = {Hung Phi Tong-Viet},
      title = {Inductive {M}c{K}ay condition in defining characteristic},
      year = {2012},
      pages = {426--438},
      }
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    @ARTICLE{Springer, mrkey = {0354894},
      issn = {0020-9910},
      author = {Springer, T. A.},
      mrclass = {20H15},
      doi = {10.1007/BF01390173},
      journal = {Invent. Math.},
      zblnumber = {0287.20043},
      volume = {25},
      mrnumber = {0354894},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {R. Steinberg},
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      pages = {159--198},
      }
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    @ARTICLE{Tits, mrkey = {0206117},
      issn = {0021-8693},
      author = {Tits, J.},
      mrclass = {20.75},
      doi = {10.1016/0021-8693(66)90053-6},
      journal = {J. Algebra},
      zblnumber = {0145.24703},
      volume = {4},
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      year = {1966},
      pages = {96--116},
      }

Authors

Gunter Malle

FB Mathematik, TU Kaiserslautern, Kaisers\-lautern, Germany

Britta Späth

Universität Wuppertal, School of Mathematics and Natural Sciences, Wuppertal, Germany