  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 $\mathrm{SL}_n$
and of $\mathrm{GL}_n$. 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
$\mathrm{GL}_n$ it also affords all the regular orbit representations as well
as the invariant characters of the Lie algebra.
