Counting p'-characters in finite reductive groups

TL;DR

This paper proves the relative McKay conjecture for certain finite reductive groups in defining characteristic, using Gelfand-Graev characters and cuspidal Levi subgroups, with explicit semisimple class counts.

Contribution

It establishes the conjecture for a broad class of groups under specific conditions and provides explicit computations of semisimple class numbers.

Findings

The relative McKay conjecture holds for G^F when certain conditions are met.

Explicit formulas for counting semisimple classes in finite reductive groups.

Verification of the conjecture for types B_n, C_n, E_6, E_7 in defining characteristic.

Abstract

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Graev characters for connected reductive groups with disconnected center developed by Digne-Lehrer-Michel and the theory of cuspidal Levi subgroups. We also explicitly compute the number of semisimple classes of G^F for any simple algebraic group G.