We consider the action of the symmetric group S n on the Schubert poly-nomials of xed degree. This action induces a well known family of representations of S n. We point on Kazhdan-Lusztig structures which appear in these representations. First, a combinatorial rule for restricting these representations to Young subgroups is obtained. This rule is an exact analogue of Barbasch-Vogan's rule for ...

In this lecture we continue to study the category O0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing projective functors Pi : O0 → O0 that act by w 7→ w(1 + si) on K0(O0). Using these functors we produce a projective generator of O0. In Section 2 we explain some of the work of Soergel that ultimately was used by Elias and Williamson to give a ...

Let %‘A be the category of finite dimensional right modules for a quasi-hereditary algebra A. In the context of various types of Kazhdan-Lusztig theories, we study both the homological dual A! = Extt, (A/ rad(A), A/ rad(A)) and the graded algebra grA = @rad(A)j/rad(A)j+‘. For example, we investigate a condition introduced in [6], and here called (SKL’), which guarantees that A!’ E gr A. A stren...

Introduction A locally compact, second countable group G is called Kazhdan if for any unitary representation of G, the first continuous cohomology group is trivial H 1 ct (G, ρ) = 0. There are several other equivalent definitions, the reader should consult [6], esp. 1.14 and 4.7. For some time now Kazhdan groups have attracted attention. One of the main challenges is to understand them geometri...

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description the -functions and Kazhdan-Lusztig cells these groups.

We use the affine Robinson–Schensted correspondence to describe structure of bidirected edges in Kazhdan–Lusztig cells type A. Equivalently, we give a comprehensive description Knuth equivalence classes permutations.

Kac and Kazhdan conjectured a character formula of a simple module with a generic highest weight at the critical level. We prove this formula for the affine Lie superalgebras with symmetrizable Cartan matrices.

We give explicit formulas for the Kazhdan-Lusztig P and Rpolynomials for permutations coming from the variety F1,n−1 of incomplete flags consisting of a line and a hyperplane.

We perform a systematic investigation of Kazhdan’s relative Property (T) for pairs (G,X), G a locally compact group and X any subset. When G is a connected Lie group or a p-adic algebraic group, we provide an explicit characterization of subsets X ⊂ G such that (G,X) has relative Property (T). In order to extend this characterization to lattices Γ ⊂ G, a notion of “resolutions” is introduced, a...

We study the parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. In particular, we show that these polynomials are always either zero or a monic power of q, and that they are combinatorial invariants.