Regular semisimple elements and involutions in finite general linear groups of odd characteristic

Chris Parker and Rob Wilson showed that involution-centraliser methods could be used for solving several problems that appeared to be computationally hard and gave complexity analyses for methods to construct involutions and their centralisers in quasisimple Lie type groups in odd characteristic. Crucial to their analyses are conjugate involution pairs whose products are regular semisimple, possibly in an induced action on a subspace. We consider the fundamental case of conjugate involution pairs, in finite general linear groups $\mathrm {GL}(n,q)$ with $q$ odd, for which the product is regular semisimple on the underlying vector space. Such involutions form essentially a single conjugacy class $\mathcal {C}$. We prove that a constant proportion of pairs from $\mathcal {C}$ have regular semisimple product. Moreover we show that for a fixed parity of $n$, this proportion converges exponentially quickly to a limit, as $n$ approaches $\infty$, the limit being $(1-q^{-1})^2\Phi (q)^3$ for even $n$ and $(1-q^{-1})\Phi (q)^3$ for odd $n$, where $\Phi (q)=\prod _{i=1}^\infty (1-q^{-i})$.