Hesselink strata in small characteristic and Lusztig-Xue pieces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912138347413504 |
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| author | Premet, Alexander |
| author_facet | Premet, Alexander |
| contents | Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p\ge 0$ and $\mathfrak{g}={\rm Lie}(G)$. We show that the nilpotent pieces ${\rm LX}(Δ)$ introduced by Lusztig coincide with the corresponding Hesselink strata $\mathcal{H}(Δ)$ and hence form a partition of the nilpotent cone of $\mathfrak{g}$. Similar results are obtained for the unipotent pieces of $G$. Here $Δ$ runs through the set of all weighted Dynkin diagrams of $G$. Thanks to the results obtained by Lusztig, Xue and Voggesberger this boils down to establishing the partition property of the pieces ${\rm LX}(Δ)$ for groups of type ${\rm E_7}$ in characteristic $2$ and for groups of type ${\rm E_8}$ in characteristic $2$ and $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hesselink strata in small characteristic and Lusztig-Xue pieces Premet, Alexander Representation Theory 14L30, 17B45 Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p\ge 0$ and $\mathfrak{g}={\rm Lie}(G)$. We show that the nilpotent pieces ${\rm LX}(Δ)$ introduced by Lusztig coincide with the corresponding Hesselink strata $\mathcal{H}(Δ)$ and hence form a partition of the nilpotent cone of $\mathfrak{g}$. Similar results are obtained for the unipotent pieces of $G$. Here $Δ$ runs through the set of all weighted Dynkin diagrams of $G$. Thanks to the results obtained by Lusztig, Xue and Voggesberger this boils down to establishing the partition property of the pieces ${\rm LX}(Δ)$ for groups of type ${\rm E_7}$ in characteristic $2$ and for groups of type ${\rm E_8}$ in characteristic $2$ and $3$. |
| title | Hesselink strata in small characteristic and Lusztig-Xue pieces |
| topic | Representation Theory 14L30, 17B45 |
| url | https://arxiv.org/abs/2407.21469 |