On generalised Deligne--Lusztig constructions

Abstract

This thesis is on the representations of connected reductive groups over finite quotients of a complete discrete valuation ring. Several aspects of higher Deligne–Lusztig representations are studied.

First we discuss some properties analogous to the finite field case; for example, we show that the higher Deligne–Lusztig inductions are compatible with the Harish-Chandra inductions.

We then introduce certain subvarieties of higher Deligne–Lusztig varieties, by taking pre-images of lower level groups along reduction maps; their constructions are motivated by efforts on computing the representation dimensions. In special cases we show that their cohomologies are closely related to the higher Deligne–Lusztig representations.

Then we turn to our main results. We show that, at even levels the higher Deligne–Lusztig representations of general linear groups coincide with certain explicitly induced representations; thus in this case we solved a problem raised by Lusztig. The generalisation of this result for a general reductive group is completed jointly with Stasinski; we also present this generalisation. Some discussions on the relations between this result and the invariant characters of finite Lie algebras are also presented.

In the even level case, we give a construction of generic character sheaves on reductive groups over rings, which are certain complexes whose associated functions are higher Deligne–Lusztig characters; they are accompanied with induction and restriction functors. By assuming some properties concerning perverse sheaves, we show that the induction and restriction functors are transitive and admit a Frobenius reciprocity.