Totally orthogonal finite simple groups

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.