Twisting operators and centralisers of Lie type groups over local rings

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 ${\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q[[\pi ]]\mathrm{/}{\pi }^r)$ on a Springer fibre intersected by Deligne–Lusztig varieties; we determine the primitivities of the induced cohomological representations for single cycles. The case of ${\mathrm{S}\mathrm{L}}_2$ over finite dual numbers is presented with a criterion on semisimple orbit representations.