Counting Parabolic Principal G-bundles with Nilpotent Sections over $\mathbb{P}^{1}$
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866916702981193728 |
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| author | Singh, Rahul |
| author_facet | Singh, Rahul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_11583 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Counting Parabolic Principal G-bundles with Nilpotent Sections over $\mathbb{P}^{1}$ Singh, Rahul Algebraic Geometry Representation Theory 14G15 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. |
| title | Counting Parabolic Principal G-bundles with Nilpotent Sections over $\mathbb{P}^{1}$ |
| topic | Algebraic Geometry Representation Theory 14G15 |
| url | https://arxiv.org/abs/2111.11583 |