  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.

Algebraic Geometry;Representation Theory