On a stability of higher level Coxeter unipotent representations
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912151218683904 |
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| author | Chen, Zhe |
| author_facet | Chen, Zhe |
| contents | Let $\mathbb{G}$ be a connected reductive group over $\mathcal{O}$, a complete discrete valuation ring with finite residue field $\mathbb{F}_q$. Let $R_{T_r,U_r}^θ$ be a level $r$ Deligne--Lusztig representation of $\mathbb{G}(\mathcal{O})$, where $r$ is a positive integer. We show that, if $q$ is not small, and if $T$ is Coxeter and $θ=1$, then $R_{T_r,U_r}^1$ degenerates to the $r=1$ case. For $\mathbb{G}=\mathrm{GL}_2$ (or $\mathrm{SL}_2$), as an application we give the dimensions and decompositions of all $R_{T_r,U_r}^θ$ for Coxeter $T$. This in turn leads us to state a conjectural sign formula for $R_{T_r,U_r}^θ$, for general $(\mathbb{G}, T, θ,r)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a stability of higher level Coxeter unipotent representations Chen, Zhe Representation Theory Let $\mathbb{G}$ be a connected reductive group over $\mathcal{O}$, a complete discrete valuation ring with finite residue field $\mathbb{F}_q$. Let $R_{T_r,U_r}^θ$ be a level $r$ Deligne--Lusztig representation of $\mathbb{G}(\mathcal{O})$, where $r$ is a positive integer. We show that, if $q$ is not small, and if $T$ is Coxeter and $θ=1$, then $R_{T_r,U_r}^1$ degenerates to the $r=1$ case. For $\mathbb{G}=\mathrm{GL}_2$ (or $\mathrm{SL}_2$), as an application we give the dimensions and decompositions of all $R_{T_r,U_r}^θ$ for Coxeter $T$. This in turn leads us to state a conjectural sign formula for $R_{T_r,U_r}^θ$, for general $(\mathbb{G}, T, θ,r)$. |
| title | On a stability of higher level Coxeter unipotent representations |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2404.00233 |