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 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.
