Counting Parabolic Principal G-Bundles with Nilpotent Sections Over \(\mathbb {P}^{1}\)

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