Abstract

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.