Counting absolutely indecomposable $G$-bundles

Abstract

For a reductive group $G$ over a finite field $k$, and a smooth projective curve $X/k$, we give a motivic counting formula for the number of absolutely indecomposable $G$-bundles on $X$. We prove that the counting can be expressed via the cohomology of the moduli stack of stable parabolic $G$-Higgs bundles on $X$. This result generalizes work of Schiffmann and work of Dobrovolska, Ginzburg, and Travkin from $\mathrm{GL}_n$ to a general reductive group. Along the way we prove some structural results on automorphism groups of $G$-torsors, and we study certain Lie-theoretic counting problems related to the case when $X$ is an elliptic curve - a case which we investigate more carefully following Fratila, Gunningham and P. Li.