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

TL;DR

This work counts parabolic principal $G$-bundles with nilpotent adjoint sections over $\mathbb{P}^1$ for split connected reductive groups $G$ over $\mathbb{F}_q$. It introduces a coproduct on parabolic data and employs a geometric approach via a Bialynicki–Birula-type stratification to reduce the triple-count problem to counting generalized Steinberg varieties, yielding explicit formulas. In the $GL_n$ case it recovers Mellit's generating function by a symmetric-function framework, including a product-factorization that ties to Levi subgroups and a global Exp-type generating function. The results generalize Mellit's vector-bundle counts to arbitrary split reductive groups, are invariant under central isogenies, and illuminate a coproduct-based route to counting parabolic bundles with nilpotent structures on $\mathbb{P}^1$.

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

Theorems & Definitions (67)

• Definition 1
• Remark 2.1
• Theorem 3.1
• Remark 3.1
• Definition 2
• Theorem 3.2
• Corollary 3.2.1
• proof
• Remark 3.2
• Corollary 3.2.2