• Geometric representation theory of classical groups (with flag varieties, quivers, affine Grassmannanians, ...)
• Algebraic combinatorics, in particular crystal bases in the sense of Kashiwara.

This paper provides a combinatorial description of Mirković-Vilonen (MV) polytopes of type A₁⁽¹⁾ and A₂⁽²⁾. Combining this construction with those contained in Affine Mirković-Vilonen polytopes (see below), we get a concrete description of MV polytopes for any symmetric affine Kac-Moody algebra g=n₋+h+n₊. This opens the way to a description of the bijections at q=0 between PBW bases of U_q(n₊).

The first goal of this paper is to study the amount of compatibility between two important constructions in the theory of quantized enveloping algebras, namely the canonical basis and the quantum Frobenius morphism. The second goal is to study orders with which the Kashiwara crystal B(∞) of a symmetrizable Kac-Moody algebra can be endowed; these orders are defined so that the transition matrices between bases naturally indexed by B(∞) are lower triangular.

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(-∞), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirković-Vilonen (MV) polytope, can be associated to each element in B(-∞). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces.

Let U be the enveloping algebra of a symmetric Kac-Moody algebra. The Weyl group acts on U, up to a sign. In addition, the positive subalgebra U⁺ contains a so-called semicanonical basis, with remarkable properties. The aim of this paper is to show that these two structures are as compatible as possible.

Let g be a simply laced semisimple complex Lie algebra. Orienting the edges of its Dynkin diagram, we get a quiver Q. The representation spaces of the preprojective algebra of Q are Lusztig's nilpotent varieties. According to Kashiwara and Saito, the set of irreducible components of these varieties provides a geometric realisation of the crystal of U⁺, the upper part of U(g). Defining reflection functors for modules over the preprojective algebra, we explain how to find the Lusztig parameters in this model. This result can be understood as the combinatorial part of the transition matrix between the semicanonical basis and the Poincaré-Birkhoff-Witt bases. Our statement is indeed more precise: we recover Anderson's and the second author's MV polytopes, and we observe that Lusztig's nilpotent varieties admit a stratification by pseudo-Weyl polytopes, in complete analogy with the GGMS strata in the affine Grassmannian.

Let G be a complex reductive group and let ^LG be its Langlands dual. Let us choose a triangular decomposition ^Ln_-+^Lh+^Ln₊ of the Lie algebra of ^LG. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannian G(C((t)))/G(C[[t]]) is a crystal isomorphic to the crystal of the canonical basis of U(^Ln₊). Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.

We propose an analogue of Solomon's descent theory for the case of a wreath product G ~ Sn, where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.