On the lifting theory of \(^ 3D_ 4(q^ 3)\) (Q1813960)

This paper, based mainly on the author's 1986 Yale Ph.D. thesis, verifies in several special cases a conjecture of \textit{N. Kawanaka} which involves ``lifting'' theory [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 851-861 (1981; Zbl 0499.20027); ibid. 30, 499-516 (1984; Zbl 0536.20023); in Proc. Symp. Pure Math. 47, 147-163 (1987; Zbl 0654.20046)]. Although the precise results are too technical to summarize here, the main point is to obtain a natural bijection between the characters of a finite group of Lie type obtained as the fixed point set of an endomorphism \(\sigma\) of the ambient algebraic group and the \(\sigma\)-fixed characters of the larger group of fixed points of \(\sigma^ m\). Here the author concentrates on subgroups of the algebraic groups \(PSO_ 8\), \(PSp_ 8\), and (especially) \(\text{Spin}_ 8\) when \(m=3\) and \(\sigma\) is the triality Frobenius map. As in Kawanaka's work, some comparisons with modular representations (in the defining characteristic) are needed.