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Last steps in a proof of the McKay Conjecture on character degrees

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GR2W02 - Simple groups, representations and applications

The talk will give an overview of what could be the endgame in the proof of McKay conjecture on character degrees. In 1972 McKay conjectured that for any prime p and any finite group G the set of irreducible characters of degree prime to p has the same cardinality in G and in the normalizer of its Sylow p-subgroup. The conjecture was reduced by Isaacs, Malle and Navarro in 2007 to a stronger statement about finite quasi-simple groups. Using the classification of the latter, this focused the effort on representations of finite groups of Lie type and in particular the determination of Irr(G) as an Out(G)-set for quasi-simple groups G and their p-local subgroups. Lusztig’s parametrization of characters of overgroups G’ corresponding to reductive groups with connected center being the main source of information, quasi-simple groups of type different from D were settled thanks to a mix of techniques including Shintani descent and generalized Gelfand-Graev representations. I will show how adding Harish Chandra theory to the picture should help settle the case of groups of type D.

This talk is part of the Isaac Newton Institute Seminar Series series.

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