Comptage de fibr\'es de Hitchin pour le groupe $\mathrm{SL}(n)$

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 $C'$ of rank $n/d$ and degree $e$ where $C'$ ranges over cyclic covers $C'$ 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 $\ell$-adic Poincar\'e polynomial and its Euler-Poincar\'e 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 $\mathrm{GL}(n)$.