On a stability of higher level Coxeter unipotent representations

Abstract

Let $\mathbb{G}$ be a connected reductive group over $\mathcal{O}$, a complete discrete valuation ring with finite residue field $\mathbb{F}_q$. Let $R_{T_r,U_r}^{\theta}$ 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 $\theta=1$, 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 $R_{T_r,U_r}^{\theta}$ for Coxeter $T$. This in turn leads us to state a conjectural sign formula for $R_{T_r,U_r}^{\theta}$, for general $(\mathbb{G}, T, \theta,r)$.