Abstract
We prove that if G is a finite simple group, then all irreducible complex representations of G may be realized over the real numbers if and only if every element of G may be written as a product of two involutions in G. This follows from our result that if q is a power of 2, then all irreducible complex representations of the orthogonal groups \(\mathrm {O}^{\pm }(2n,\mathbb {F}_q)\) may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups.
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Acknowledgements
The author thanks Mandi Schaeffer Fry and Jay Taylor for helpful correspondence regarding Proposition 6.1. The author was supported in part by a grant from the Simons Foundation, Award # 280496.
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