Comptage de fibrés de Hitchin pour le groupe $\mathrm{S}\mathrm{L}(n)$

Let $C$ be a smooth projective curve of genus $g$ over a finite field ${\mathbb{F}}_q$ and let $D$ be a divisor on $C$ of degree $>2g-2$. We assume that the characteristic of ${\mathbb{F}}_q$ is sufficiently large. Let $n$ be an integer and let $\beta$ be a line bundle on $C$ of degree $e$, coprime to $n$. We give a formula for the number of stable ($D$-twisted) Hitchin bundles over $C$ of rank $n$ and determinant $\beta$ in terms of the number of stable Hitchin bundles over $C^{\mathrm{\prime }}$ of rank $n\mathrm{/}d$ and degree $e$ where $C^{\mathrm{\prime }}$ ranges over cyclic covers $C^{\mathrm{\prime }}$ of $C$ of degree $d$ dividing $n$. Using a work by Mozgovoy-O’Gorman, we derive a closed formula for the following invariants of the moduli space of ($D$-twisted) Hitchin bundles over $C$ of rank $n$, trace $0$ and determinant $\beta$: its number of points over finite extensions of ${\mathbb{F}}_q$, its $\mathrm{\ell }$-adic Poincaré polynomial and its Euler-Poincaré characteristic. Our main tools are the fundamental lemma of automorphic induction and a support theorem for the relative cohomology of a local system on the Hitchin fibration for the group $\mathrm{G}\mathrm{L}(n)$.