Some Reality Properties of Finite Simple Orthogonal Groups

Abstract

We prove several reality properties for finite simple orthogonal groups. For any prime power q and m ≥ 1, we show that all real conjugacy classes are strongly real in the simple groups PΩ ± (4m+2, q), m ≥ 1, except in the case PΩ − (4m+2, q) with q ≡ 3(mod 4), and we construct weakly real classes in this exceptional case for any m. We also show that no irreducible complex character of PΩ ± (n, q) can have Frobenius-Schur indicator −1, except possibly in the case PΩ − (4m + 2, q) with q ≡ 3(mod 4).