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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 anexplicit formula for the number of $\mathbb{F}_{q}$-rational points ofgeneralized Steinberg varieties of $G$. Secondly, for each principal $G$-bundleover $\mathbb{P}^1$, we give an explicit formula counting the number of triplesconsisting of parabolic structures at $0$ and $\infty$ and a compatiblenilpotent section of the associated adjoint bundle. In the case of $GL_{n}$ wecalculate a generating function of such volumes re-deriving a result of Mellit.

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