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The McKay conjecture for exceptional groups and odd primes

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Abstract

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G F, such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C G (S), the centraliser of S in G, extend to their inertia group in N G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14].

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  1. Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, 52056, Aachen, Germany

    Britta Späth

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  1. Britta Späth
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Correspondence to Britta Späth.

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This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.

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Späth, B. The McKay conjecture for exceptional groups and odd primes. Math. Z. 261, 571–595 (2009). https://doi.org/10.1007/s00209-008-0340-7

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