We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GLn(q). We show that the geometric version of Broué’s conjecture over Q , together with Craven’s formula, holds for any unipotent block whenever it holds for the principal Φ1-block.

Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. After presenting some consequences of this important theorem, these lectures explain the main ideas of the proof. Some algebraic technicalities will be pushed to the background. Chapter 1 is the main part of the bo...

We give a new geometric construction of the big projective module in the principal block of the BGG category O, or rather the corresponding D-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheave...

Let G be an affine algebraic group over an algebraically closed field whose identity component G0 is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G0 is unipotent. In this paper we define a map from the set of “twisted conjugacy classes” in W to the set of unipotent G0-conjugacy classes in D generalizing an earlier construction which applied wh...

We consider the discrete representations of 3-manifold groups into PU(2, 1) that appear in the Falbel-Koseleff-Rouillier, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. This illustrates the fact that a discrete representation π1(M) → PU(2, 1) with cyclic u...

Let G be a connected reductive algebraic group over the algebraic closure of a finite field Fq of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(Fq). In particular, we show that in a suitable sense, “large” portions of these two assignments in fact coincide. This extends...

The aim of this note is to classify all weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that 2 dimX + dim cg(X) < dim g for X a non-trivial unipotent subgroup of the connected simple algebraic group G. This shows the failure of the analogue of the so called “2F-condition” for finite groups for the adjoi...

In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a nite reductive group G(F q ). This is de ned in terms of certain average values of the irreducible characters on unipotent classes. The problem was solved by Lusztig [16] for the case where q is a power of a su ciently large prime. In this paper we show that, in general, these ...

0. Introduction 1105 1. Constructible families 1111 2. Genericity for finite subgroups 1116 3. Finite groups of Lie type 1119 4. Basic nonconcentration estimate 1122 5. Finite subgroups of abelian varieties 1126 6. Orders of conjugacy classes and centralizers 1127 7. Regular semisimple and unipotent elements 1129 8. Minimal unipotent elements 1134 9. Frobenius map 1140 10. Traces in the basic c...