Twisting operators and centralisers of Lie type groups over local rings

TL;DR

This paper extends classical results on twisting operators in the context of Lie type groups over local rings, exploring properties of centralisers and constructing actions on Springer fibres with applications to cohomological representations.

Contribution

It introduces new constructions of group actions on Springer fibres over local rings and analyzes the primitivity of associated cohomological representations, extending classical character value preservation results.

Findings

Twisting operators preserve certain Deligne--Lusztig character values.

Constructed an action of $ ext{GL}_n( ext{local ring})$ on Springer fibres intersected with Deligne--Lusztig varieties.

Determined primitivity conditions for cohomological representations in specific cases.

Abstract

We extend the classical result asserting that the twisting operator preserves certain Deligne--Lusztig character values for truncated formal power series; along the way we discuss some properties of centralisers. This leads us to the construction of an action of $\mathrm{GL}_n(\mathbb{F}_q[[\pi]]/\pi^r)$ on a Springer fibre intersected by Deligne--Lusztig varieties; we determine the primitivities of the induced cohomological representations for single cycles. The case of $\mathrm{SL}_2$ over finite dual numbers is presented with a criterion on semisimple orbit representations.