On a stability of higher level Coxeter unipotent representations

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 $\mathbb{G}={\mathrm{G}\mathrm{L}}_2$ (or ${\mathrm{S}\mathrm{L}}_2$), 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 $(\mathbb{G},T,\theta ,r)$.