Representations of Reductive Groups over Quotients of Local Rings

Abstract

In some recent work, Lusztig outlined a generalisation of the construction of Deligne and Lusztig to reductive groups over finite rings coming from the ring of integers in a local field, modulo some power of the maximal ideal. Lusztig conjectures that all irreducible representations of these groups are contained in the cohomology of a certain family of varieties. We show that, contrary to what was expected, there exist representations that cannot be realised by the varieties given by Lusztig. Moreover, we show how the remaining representations in the case under consideration can be realised in the cohomology of a different kind of variety. This may suggest a way to reformulate Lusztig's conjecture.