Character degree sums and real representations of finite classical groups of odd characteristic

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\mathrm{G}\mathrm{S}\mathrm{p}(2n,{\mathbb{F}}_q)$ and ${\mathrm{G}\mathrm{O}}^{\pm }(2n,{\mathbb{F}}_q)$ denote the symplectic and orthogonal groups of similitudes over ${\mathbb{F}}_q$, respectively. We prove that every real-valued irreducible character of $\mathrm{G}\mathrm{S}\mathrm{p}(2n,{\mathbb{F}}_q)$ or ${\mathrm{G}\mathrm{O}}^{\pm }(2n,{\mathbb{F}}_q)$ 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 $\mathit{G}$ is a classical connected group defined over ${\mathbb{F}}_q$ with connected center, with dimension $d$ and rank $r$, then the sum of the degrees of the irreducible characters of $\mathit{G}({\mathbb{F}}_q)$ is bounded above by $(q+1{)}^{(d+r)\mathrm{/}2}$. Finally, we show that if $\mathit{G}$ is any connected reductive group defined over ${\mathbb{F}}_q$, for any $q$, the sum of the degrees of the irreducible characters of $\mathit{G}({\text{FF}}_q)$ is bounded below by $q^{(d-r)\mathrm{/}2}(q-1{)}^r$. We conjecture that this sum can always be bounded above by $q^{(d-r)\mathrm{/}2}(q+1{)}^r$.