Character Degree Sums and Real Representations of Finite Classical Groups of Odd Characteristic

Let Fq be a finite field with q elements, where q is the power of an odd prime, and let GSp(2n, Fq) and GO±(2n, Fq) denote the symplectic and orthogonal groups of similitudes over Fq, respectively. We prove that every real-valued irreducible character of GSp(2n, Fq) or GO±(2n, Fq) is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if G is a classical connected group defined over Fq with connected center, with dimension d and rank r, then the sum of the degrees of the irreducible characters of G(Fq) is bounded above by (q + 1)(d+r)/2. Finally, we show that if G is any connected reductive group defined over Fq, for any q, the sum of the degrees of the irreducible characters of G(Fq) is bounded below by q(d−r)/2(q − 1)r . We conjecture that this sum can always be bounded above by q(d−r)/2(q + 1)r .