Dade's invariant conjecture for Steinberg's triality groups [formula omitted] in defining characteristic

Abstract

In this paper, we verify Dade's invariant conjecture for Steinberg's triality groups D 4 3 ( 2 n ) in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade's conjecture for Steinberg triality groups D 4 3 ( q ) in non-defining characteristics, Math. Z. 241 (2002) 445–469] and [J. An, F. Himstedt, S. Huang, Uno's invariant conjecture for Steinberg's triality groups in defining characteristic, in preparation], this completes the proof of Dade's conjecture for Steinberg's triality groups. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for D 4 3 ( 2 n ) in the defining characteristic, i.e., D 4 3 ( 2 n ) is good for the prime 2 in the sense of Isaacs, Malle and Navarro.