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 $(\mathbb{ G}, T, \theta,r)$.
Categories: math.RT