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

TL;DR

This paper provides explicit formulas for counting rational points on generalized Steinberg varieties and for enumerating parabolic principal G-bundles with nilpotent sections over the projective line, extending known results.

Contribution

It introduces explicit formulas for counting rational points on Steinberg varieties and for parabolic G-bundles with nilpotent sections, including a generating function for GL_n.

Findings

Explicit formula for the number of rational points on Steinberg varieties.

Counting formula for parabolic G-bundles with nilpotent sections over P^1.

A generating function for volumes in the case of GL_n.

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 $GL_{n}$ we calculate a generating function of such volumes re-deriving a result of Mellit.