For some time we have studied the representation theory of semisimple algebraic groups in characteristic p > 0. Of special interest is a celebrated conjecture of Lusztig [Ll] describing the characters of the simple modules when p is not too small relative to the root system. A similar conjecture, by Kazhdan and Lusztig [KLl], for the composition factor multiplicities of Verma modules for semisi...

We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and quantum cohomology complex flag variety. The main theorem provides combinatorial formula for unique maximum element in this poset terms paths Bruhat graph, whose vertices are indexed by elements finite Weyl group. Key to establishing connection is fact that graph e...

We study the abelian category W (p)-mod of modules over the triplet W algebra W (p). We construct the projective covers P± s of all the simple objects X± s , 1 ≦ s ≦ p, in the category W (p)-mod. By using the structure of these projective modules, we show that W (p)-mod is a category which is equivalent to the abelian category of the finite-dimensional modules for the restricted quantum group Ū...

We will construct the Lusztig form for the quantum loop algebra of gln by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine gln by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canoni...

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan–Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert c...

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit c...

Using Du’s characterization of the dual canonical basis of the coordinate ring O(GL(n,C)), we express all elements of this basis in terms of immanants. We then give a new factorization of permutations w avoiding the patterns 3412 and 4231, which in turn yields a factorization of the corresponding Kazhdan-Lusztig basis elements C w(q) of the Hecke algebra Hn(q). Using the immanant and factorizat...

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BCn and An; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BCn, we also consider their “Type II” integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint inte...

In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig in the case where the characteristic of the field over which the group is defined is large enough. The first named author extended this to the case where the characteristic is good. It is the purpose of this paper to ...