Dadeʹs invariant conjecture for Steinbergʹs triality groups in defining characteristic

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