Counting Parabolic Principal G-bundles with Nilpotent Sections over ${\mathbb{P}}^1$

Abstract

Let $G$ be a split connected reductive group over ${\mathbb{F}}_q$ and let ${\mathbb{P}}^1$ be the projective line over ${\mathbb{F}}_q$. Firstly, we give an explicit formula for the number of ${\mathbb{F}}_q$-rational points of generalized Steinberg varieties of $G$. Secondly, for each principal $G$-bundle over ${\mathbb{P}}^1$, we give an explicit formula counting the number of triples consisting of parabolic structures at $0$ and $\infty$ and a compatible nilpotent section of the associated adjoint bundle. In the case of $G{L}_n$ we calculate a generating function of such volumes re-deriving a result of Mellit.

Final version accepted for publication in Transformation Groups