# Flags and orbits of connected reductive groups over local rings

## TL;DR

This paper explores the structure of higher Deligne-Lusztig representations over local rings, establishing non-nilpotency conditions, orbit relations, and a combinatorial construction that generalizes existing representations for linear groups.

## Contribution

It introduces a combinatorial analogue of Deligne-Lusztig construction for linear groups over local rings, encompassing all nilpotent and regular orbit representations.

## Key findings

- Higher Deligne-Lusztig representations are non-nilpotent under certain conditions.
- Relations between orbits of representations of SL_n and GL_n are established.
- A combinatorial construction generalizes existing representations and includes all nilpotent and regular orbit representations.

## 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.