Some Reality Properties of Finite Simple Orthogonal Groups

We prove several reality properties for finite simple orthogonal groups. For any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}Ω^{\pm}(4m+2,q), m \geq 1$, except in the case $\mathrm{P}Ω^{-}(4m+2,q)$ 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 $\mathrm{P}Ω^{\pm}(n,q)$ can have Frobenius-Schur indicator $-1$, except possibly in the case $\mathrm{P}Ω^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$.

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