Flags and orbits of connected reductive groups over local rings

Abstract

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.