Flags and orbits of connected reductive groups over local rings

We prove that generic higher Deligne-Lusztig representations over truncated formal power series are non-nilpotent, when the parameters are non-trivial on the biggest reduction kernel of the centre; we also establish a relation between the orbits of higher Deligne-Lusztig representations of ${\mathrm{S}\mathrm{L}}_n$ and of ${\mathrm{G}\mathrm{L}}_n$. Then we introduce a combinatorial analogue of Deligne-Lusztig construction for general and special linear groups over local rings; this construction generalises the higher Deligne–Lusztig representations and affords all the nilpotent orbit representations, and for ${\mathrm{G}\mathrm{L}}_n$ it also affords all the regular orbit representations as well as the invariant characters of the Lie algebra.