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 SL n and of 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 GL n it also affords all the regular orbit representations as well as the invariant characters of the Lie algebra.

Twisting operators and centralisers of Lie type groups over local rings

“…Recently, this work has been studied and generalised along various aspects; see e.g. [Sta09], [Sta11], [Che18b], [Che20], [CI19]. In this section we study the representations of G(O r ) via Deligne-Lusztig theory in two directions, one through the twisting operators and one through the Springer fibres.…”

The Drinfeld stratification for $${{\,\mathrm{GL}\,}}_n$$

“…In the Appendix, we present an analysis of the fibers of the natural projection maps X h → X h−1 ; we believe this could be a possible approach to proving Conjecture 7.2.1 and may be of independent interest. It would be interesting to see if the Drinfeld stratification plays a role in connections to orbits in finite Lie algebras, à la work of Chen [Che19].…”

Intersections of Deligne--Lusztig varieties and Springer fibres

“…Indeed, if u ∈ U F is rectangular, then the finite quotient group GL d (F q [[π]]/π r ) acts on B u,w because GL d (F q [[π]]/π r ) is naturally isomorphic to the G-centraliser of where A i ∈ M d (F q ). (See [Che20a,5.3] and [Che20b,4.6]; note that the notation used there is different up to a blocked transpose.) Thus we get a virtual representation…”