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

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.