We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type An−1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for ...

In this paper we study the partially ordered set Q of cells in Rietsch’s [20] cell decomposition of the totally nonnegative part of an arbitrary flag variety P ≥0 . Our goal is to understand the geometry of P ≥0 : Lusztig [13] has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P ≥0 is homeomorphic to a ball. The order comp...

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 ...

Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group G, including Bernstein’s presentation and description of the center, Macdonald’s formula, the CasselmanShalika formula, and the Lusztig-Kato formula. There are no new results here, and the same is essentially true of the proofs. We have been strongly influence...

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 ...

We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in [21] in terms of ribbon tableaux.

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.

We provide a short proof on the change-of-basis coefficients from Specht basis to Kazhdan-Lusztig basis, using theory for parabolic Hecke algebra.