Counting absolutely indecomposable $G$-bundles

TL;DR

This paper provides a motivic counting formula for absolutely indecomposable G-bundles on a curve over a finite field, linking it to the cohomology of moduli stacks of stable parabolic G-Higgs bundles, extending previous work from GL_n to general reductive groups.

Contribution

It generalizes existing counting formulas from GL_n to all reductive groups and establishes a connection with the cohomology of moduli stacks of G-Higgs bundles.

Findings

Derived a motivic counting formula for G-bundles

Connected counting to cohomology of moduli stacks

Explored automorphism groups and Lie-theoretic problems

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.