Joint distribution of inverses in matrix groups over finite fields

We study the joint distribution of the solutions to the equation $gh=x$ in $G({\mathbb{F}}_p)$ as $p\to \mathrm{\infty }$, for any fixed $x\in G(\mathbb{Z})$, where $G={\mathrm{GL}}_n$, ${\mathrm{SL}}_n$, ${\mathrm{Sp}}_{2n}$ or ${\mathrm{SO}}_n^{\pm }$. In the special linear case, this answers in particular a question raised by S. Hu and Y. Li, and improves their error terms. Similar results are derived in certain subgroups, and when the entries of $g,h$ lie in fixed intervals. The latter shows for example the existence of $g\in {\mathrm{GL}}_n({\mathbb{F}}_p)$ such that $g,g^{-1}$ have all entries in $[0,c_n{p}^{1-1\mathrm{/}(2n^2+2)+\epsilon }]$ for some absolute constant $c_n>0$. The key for these results is to use Deligne’s extension of the Weil conjectures on a sheaf on $G$, along with the stratification theorem of Fouvry, Katz and Laumon, instead of reducing to bounds on classical Kloosterman sums.