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 ℓ such that a Sylow ℓ-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 they only lie in very particular 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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Abstract. We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture.

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type A n−1 , split or twisted. Key to the proofs is the study of certain characters of SL n (q) and SU n (q) related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.1991 Mathematics Subject Classification. 20C15,20C25,20C33,20D06,20D20.

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