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 $m\ge 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}{\Omega }^{\pm }(4m+2,q),m\ge 1$, except in the case $\mathrm{P}{\Omega }^{-}(4m+2,q)$ with $q\equiv 3(\mathrm{mod}4)$, and we construct weakly real classes in this exceptional case for any $m$. We also show that no irreducible complex character of $\mathrm{P}{\Omega }^{\pm }(n,q)$ can have Frobenius-Schur indicator $-1$, except possibly in the case $\mathrm{P}{\Omega }^{-}(4m+2,q)$ with $q\equiv 3(\mathrm{mod}4)$.

21 pages