Equivariant character correspondences and inductive McKay condition for type A

Abstract

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.

Descent equalities and the inductive McKay condition for types B and E

“…On that group, C g acts by diagonal automorphisms, since C does so on G 2 = g G F 2 . Our statement then follows from Theorem 3.10 and [CS17a,Thm. 4.1].…”

“…The celebrated McKay conjecture on character degrees asserts that for any finite group G and prime p |Irr p ′ (G)| = |Irr p ′ (N G (P ))| (1.0.1) where Irr p ′ denotes the set of irreducible characters of degree prime to p and P is a Sylow p-subgroup of G. The reduction theorem proved by Isaacs-Malle-Navarro [IMN07] reduces this conjecture to the checking of the so-called inductive McKay condition for each finite simple group S and prime p (see [IMN07,§10]). The inductive McKay condition has been checked for many simple groups leaving open the cases of simple groups of Lie types B, D, 2 D, E 6 , 2 E 6 , E 7 for odd p and different from the characteristic of the group, see [Ma08a], [S12], [CS13], [MS16], [CS17a], [CS17b]. One of the main results of the present paper is the following.…”

“…Proof. Note that the cases where G F sc / Z(G F sc ) is a simple group but Z(G F sc ) is smaller than its Schur multiplier reduce to the same verification as in the other cases, thus dealing only with G F sc , thanks to [Ma08a,4.1] and [CS17a,7.4]. So we will check the inductive McKay condition of [IMN07] with G F sc acting as the universal covering of G F sc / Z(G F sc ).…”

“…In the proof the norm map introduced by Shintani [Shi76] is used in an elementary and effective way. Those general statements are probably of independent interest with possible applications to the Jordan decomposition of characters, as in [CS17a,§8]. Given how central McKay's conjecture has proved to be in representation theory of finite groups, it is not surprising that the property of stabilizers A(∞) has applications to the checking of other counting conjectures through the classification of finite simple groups (see for instance [CS15]).…”

The McKay Conjecture for Finite Groups with Abelian Sylow 3-Subgroups

“…For simple groups of Lie type, the inductive McKay condition has been verified for the defining characteristics in [11]. For primes different from the defining characteristics, several series of simple groups of Lie type have been proved to be McKay-good, such as groups of type A (see [3]), Suzuki groups (see [5]), Ree groups (see [2,5]) and some series of exceptional groups. Recently, Malle and Späth have proved the McKay conjecture for ℓ = 2 via the inductive McKay condition in [9].…”