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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.00233 |
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Table of 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 $(\mathbb{G}, T, θ,r)$.