Twisting operators and centralisers of Lie type groups over local rings

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 GL n (F q [[π]]/π 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 SL 2 over finite dual numbers is presented with a criterion on semisimple orbit representations.

Intersections of Deligne--Lusztig varieties and Springer fibres

“…Indeed recently we found that, for a specific unipotent u, the intersection B u,w := X w ∩ B u appears in the study of smooth representations of the profinite group GL d (F q [[π]]), where d is a divisor of n (see [Che20b]). This serves the initial motivation for our attention on the relations between these two varieties.…”

“…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…”