Kazhdan-Lusztig Cells and the Murphy Basis

Let H be the Iwahori–Hecke algebra associated with S_n,the symmetric group on n symbols. This algebra has two importantbases: the Kazhdan–Lusztig basis and the Murphy basis.We establish a precise connection between the two bases, allowingus to give, for the first time, purely algebraic proofs fora number of fundamental properties of the Kazhdan–Lusztigbasis and Lusztig's results on the a-function. 2000 MathematicsSubject Classification 20C08.     

On matrix coefficients of the Satake isomorphism: complements to the paper of M. Rapoport

We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan–Lusztig polynomials and to Bernstein functions. Received: 29 June 1999 / Revised version: 7 September 1999     

Andersen Filtration and Hard Lefschetz

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category we define a natural filtration following Andersen [A] and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan–Lusztig polynomials. Received: May 2006, Revision: July 2007, Accepted: July 2007     

On the Induction of Kazhdan-Lusztig Cells

Barbasch and Vogan showed that the Kazhdan–Lusztig cellsof a finite Weyl group are compatible with parabolic subgroups.Their proof uses the known bridge between the theory of cellsand the theory of primitive ideals. In this paper, an elementary,self-contained proof of this result is provided, which worksfor arbitrary Coxeter groups and Lusztig's general definitionof cells (involving Iwahori–Hecke algebras with unequalparameters). The argument is based on a recent paper by Howlettand Yin. 2000 Mathematics Subject Classification 20C08.     

The intersection cohomology of Schubert varieties is a combinatorial invariant

We give an explicit and entirely poset-theoretic way to compute, for any permutation v, all the Kazhdan–Lusztig polynomials P_x,y for x,y≤v, starting from the Bruhat interval [e,v] as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells.     

Standard Modules for Tabular Algebras

We introduce cell modules for the tabular algebras defined in a previous work; these modules are analogous to the representations arising from left Kazhdan–Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type A equipped with its Kazhdan–Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.     

Monomials and Temperley–Lieb Algebras

We classify the “fully tight” simply laced Coxeter groups, that is, the ones whoseiji-avoiding Kazhdan–Lusztig basis elements are monomials in the generatorsB_si. We then investigate the basis of the Temperley–Lieb algebra arising from the Kazhdan–Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.     

On 321-Avoiding Permutations in Affine Weyl Groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A _{n – 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 a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GL _n ()     

The degenerate analogue of Ariki’s categorification theorem

We explain how to deduce the degenerate analogue of Ariki’s categorification theorem over the ground field \mathbbC{\mathbb{C}} as an application of Schur–Weyl duality for higher levels and the Kazhdan–Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

We study the parabolic Kazhdan–Lusztig polynomials for the quasi-minuscule quotients of Weyl groups. We give explicit closed combinatorial formulas for the parabolic Kazhdan–Lusztig polynomials of type q. Our study implies that these are always either zero or a monic power of q, and that they are not combinatorial invariants. We conjecture a combinatorial interpretation for the parabolic Kazhdan–Lusztig polynomials of type −1.     

On the determination of Kazhdan–Lusztig cells for affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells which have important applications in representation theory. We study the case where W is an affine Weyl group of type . Using explicit computation with COXETER and CHEVIE, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. A key ingredient of the proof is a general result which shows that the Kazhdan–Lusztig polynomials of affine Weyl group are invariant under (large enough) translations.     

Leading coefficients of Kazhdan–Lusztig polynomials and fully commutative elements

Let W be a Coxeter group of type . We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P _x,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).     

Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

Simultaneous prime specializations of polynomials over finite fields

Recently the author proposed a uniform analogue of the Bateman–Hornconjectures for polynomials with coefficients from a finitefield (that is, for polynomials in F_q[T] rather than Z[T]).Here we use an explicit form of the Chebotarev density theoremover function fields to prove this conjecture in particularranges of the parameters. We give some applications includingthe solution of a problem posed by Hall.     

Linear Spaces on Cubic Hypersurfaces, and Pairs of Homogeneous Cubic Equations

A remarkable theorem of Birch [2] shows that a system of homogeneouspolynomials with rational coefficients has a non-trivial zero,provided only that these polynomials are of odd degree, andthe system has sufficiently many variables in terms of the numberand degrees of these polynomials. Despite four decades of effort,the problem of obtaining a reasonable bound for the latter numberof variables has proved to be one of great difficulty. Whenthe system consists of a single cubic form, Davenport [4] hassucceeded in showing that 16 variables suffice, and Schmidt[17, 18, 19, 20] has devoted a series of papers to systems ofcubic forms, showing in particular that 5140 variables sufficefor pairs of cubic forms, and that (10r)⁵ variables sufficefor systems of r cubic forms. The current state of knowledgefor forms of higher degree is, by comparison, extremely weak(but see [21, 22]), and so it seems worthwhile expending furthereffort on the case of systems of cubic forms. In this paperwe improve on Schmidt's result for pairs of cubic forms. Incontrast with the sophisticated versions of the Hardy–Littlewoodmethod employed by Davenport and Schmidt, our approach is basedon an elementary idea of Lewis [12], and is applicable in arbitrarynumber fields. This method also has consequences for the existenceof linear spaces of rational solutions on cubic hypersurfaces,thereby improving on work of Lewis and Schulze-Pillot [14] onthis topic. 1991 Mathematics Subject Classification 11D72, 11E76.     

On the Gaussian Curvature of Maximal Surfaces and the Calabi-Bernstein Theorem

In this paper, a new approach to the Calabi–Bernsteintheorem on maximal surfaces in the Lorentz–Minkowski spaceL³ is introduced. The approach is based on an upper bound forthe total curvature of geodesic discs in a maximal surface inL³, involving the local geometry of the surface and its hyperbolicimage. As an application of this, a new proof of the Calabi–Bernsteintheorem is provided.