Let $G$ be a connected split reductive group over $mathbb{Z}$. Let $n$ be a positive integer. Let $i_n(q)$ be the number of elements of $G(mathbb{F}_q)$ satisfying $x^n=1$.

Question: Is there a “nice” case-free formula for $i_n(q)$? Is it always a polynomial in $q$?

The “best hope” is a formula akin to Steinberg’s formula for the number of elements of $G(mathbb{F}_q)$ (cf. Section 7.3 of these notes). Can this best hope be realised?

For instance, suppose $n=2$. Then we are considering the number of involutions in finite reductive groups. A case by case analysis shows that there exists polynomials $P, Rin mathbb{Z}(t)$ such that $i_2(q)=P(q)$ for all even $q$ and $i_2(q)=R(q)$ for all odd $q$, cf. this paper. However, I do not know a case-free proof of this fact. Nor do I know a case free expression for these polynomials.