Dade's invariant conjecture for Steinberg's triality groups in defining characteristic

In this paper, we verify Dade's invariant conjecture for Steinberg's triality groups in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade's conjecture for Steinberg triality groups 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 in the defining characteristic, i.e., is good for the prime 2 in the sense of Isaacs, Malle and Navarro.