# gr.group theory – Number of involutions in finite reductive groups

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.