Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomologica...

General facts of linear algebra are used to give proofs for the (wellknown) existence of analogs of Kazhdan-Lusztig polynomials corresponding to formal analogs of the Kazhdan-Lusztig involution, and of explicit formulae (some new, some known) for their coefficients in terms of coefficients of other natural families of polynomials (such as the corresponding formal analogs of the Kazhdan-Lusztig ...

In a seminal 1994 paper [20], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P )≥0 of an arbitrary (generalized, partial) flag variety G/P . He referred to this space as a “remarkable polyhedral subspace,” and conjectured a decomposition into cells, which was subsequently proven by the first author [27]. In this article we use discrete Morse theo...

Part 1. Two analogues of Deligne–Lusztig varieties for p-adic groups 5 2. Affine Deligne–Lusztig varieties at infinite level 5 2.1. Preliminaries 5 2.2. Deligne–Lusztig sets/varieties 7 2.3. Affine Deligne–Lusztig varieties and covers 8 2.4. Scheme structure 9 3. Case G = GLn, b basic, w Coxeter 12 3.1. Notation 12 3.2. Basic σ-conjucacy classes. Isocrystals 12 3.3. The admissible subset of Vb ...

In this paper we prove the fundamental lemma for Deligne–Lusztig functions. Namely, for every Deligne–Lusztig function φ on a p-adic group G we write down an explicit linear combination φ of Deligne–Lusztig functions on an endoscopic group H such that φ and φ have “matching orbital integrals”. In particular, we prove a conjecture of Kottwitz [Ko4]. More precisely, we do it under some mild restr...

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1, . . . , xn,n] to construct irreducible Sn-modules. This construction produces exactly the same matrices as the Kazhdan-Lusztig construction [Invent.Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. It also produces exactly the same modules as those which Clausen constructed using a different basis ...

Let (W,S) be a Coxeter system. A W -graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the W -graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis as well as this graph’s strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig g...

We introduce a new multiplication in the incidence algebra of a partially ordered set, and study the resulting algebra. As an application of the properties of this algebra we obtain a combinatorial formula for the Kazhdan-Lusztig-Stanley functions of a poset. As special cases this yields new combinatorial formulas for the parabolic and inverse parabolic Kazhdan-Lusztig polyno-mials, for the gen...

We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan–Lusztig cells using a canonical basis for a generalized version of the Temperley–Lieb algebra. Cellules pleinement commutatives de Kazhdan–Lusztig Nousétudions la compatibilité entre l'ensemble deséléments pleinement commu-tatifs d'un groupe de Coxeter et les divers typ...