2021/09/16 by Späth, Britta · 3 citations
#20C20 (20C33 20C34) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2109.08230
This is a contribution to the study of \operatorname Irr(G) as an \operatorname Aut(G)-set for G a finite quasi-simple group. Focusing on the last open case of groups of Lie type \mathrm D and 2\mathrm D, a crucial property is the so-called condition A'(∞) expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in \operatorname Irr(G). This is part of the stronger A(∞) condition introduced in the context of the reduction of the McKay conjecture to a question on quasi-simple groups. Our main theorem is that a minimal counter-example to condition A(∞) for groups of type \mathrm D would still satisfy A'(∞). This will be used in a second paper to fully establish A(∞) for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of arbitrary standard Levi subgroups of G=\mathrm D l,sc(q) extend to their stabilizers in the normalizer of that Levi subgroup. This allows to control the action of automorphisms on these extensions. From there Harish Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.