The action of component groups on irreducible components of Springer fibers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914063088353280 |
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| author | Hoang, Do Kien |
| author_facet | Hoang, Do Kien |
| contents | Let $G$ be a simple Lie group. Consider a nilpotent element $e\in \mathfrak{g}$. Let $Z_G(e)$ be the centralizer of $e$ in $G$, and let $A_e:= Z_G(e)/Z_G(e)^{o}$ be its component group. Write $\text{Irr}(\mathcal{B}_e)$ for the set of irreducible components of the Springer fiber $\mathcal{B}_e$. We have an action of $A_e$ on $\text{Irr}(\mathcal{B}_e)$. When $\mathfrak{g}$ is exceptional, we give an explicit description of $\text{Irr}(\mathcal{B}_e)$ as an $A_e$-set. For $\mathfrak{g}$ of classical type, we describe the stabilizers for the $A_e$-action. With this description, we prove a conjecture of Lusztig and Sommers. These results suggest relations (first proposed by Lusztig) between Springer fibers and cells in Weyl groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The action of component groups on irreducible components of Springer fibers Hoang, Do Kien Representation Theory Algebraic Geometry Combinatorics Let $G$ be a simple Lie group. Consider a nilpotent element $e\in \mathfrak{g}$. Let $Z_G(e)$ be the centralizer of $e$ in $G$, and let $A_e:= Z_G(e)/Z_G(e)^{o}$ be its component group. Write $\text{Irr}(\mathcal{B}_e)$ for the set of irreducible components of the Springer fiber $\mathcal{B}_e$. We have an action of $A_e$ on $\text{Irr}(\mathcal{B}_e)$. When $\mathfrak{g}$ is exceptional, we give an explicit description of $\text{Irr}(\mathcal{B}_e)$ as an $A_e$-set. For $\mathfrak{g}$ of classical type, we describe the stabilizers for the $A_e$-action. With this description, we prove a conjecture of Lusztig and Sommers. These results suggest relations (first proposed by Lusztig) between Springer fibers and cells in Weyl groups. |
| title | The action of component groups on irreducible components of Springer fibers |
| topic | Representation Theory Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2409.04076 |