Number of cuspidal automorphic representations and Hitchin's moduli spaces

Let $F$ be the function field of a projective smooth geometrically connected curve $X$ defined over a finite field $\mathbb{F}_q$. Let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Let $S$ be a non-empty finite set of points of $X$. We are interested in the number of $G$ cuspidal automorphic representations whose local behaviors in $S$ are prescribed. In this article, we consider those cuspidal automorphic representations whose local component at each $v\in S$ contains a fixed irreducible Deligne-Lusztig induced representation of a hyperspecial group. We express that the count in terms of groupoid cardinality of $\mathbb{F}_q$-points of Hitchin moduli stacks of groups associated with $G$. In the course of the proof, we study the geometry of Hitchin moduli stacks and prove some vanishing results on the geometric side of a variant of the Arthur-Selberg trace formula for test functions with small support.