On a stability of higher level Coxeter unipotent representations

Let G be a connected reductive group over O, a complete discrete valuation ring with finite residue field Fq. Let RTᵣ,Uᵣθ be a level r Deligne--Lusztig representation of G(O). We show that, if q is not small, the Coxeter unipotent RTᵣ,Uᵣ¹ degenerates to the $r=1$ case. For G=GL₂ (or SL₂), as an application we give the dimensions and decompositions of all Coxeter RTᵣ,Uᵣθ. For general G we state a conjectural sign formula for RTᵣ,Uᵣθ.