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Mirkovi\'c-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal $B(\infty)$ of a semisimple group $G$. We retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr O(N)$ of regular functions on a maximal unipotent subgroup $N$ of $G$. We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirkovi\'c-Vilonen) of $\mathscr O(N)$ manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras., Comment: 30 pages

Let G be a reductive connected algebraic group over the field of complex numbers. Through the geometric Satake equivalence, the fundamental classes of the Mirkovi\'c-Vilonen cycles define a basis in each tensor product of rational irreducible representations of G. In the case G=SL_2(C), we show that this basis coincides with the dual canonical basis at q=1., Comment: 31 pages

The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a complex connected reductive group G. In their study of this correspondence, Mirkovi\'c and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig., Comment: 77 pages

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6., Comment: 82 pages, with an appendix by Calder Morton-Ferguson and Anne Dranowski

These notes are devoted to a detailed exposition of the proof of the Geometric Satake Equivalence for general coefficients, following Mirkovic-Vilonen.

For every finite Coxeter group W , we prove that the number of chains in the shard intersection lattice introduced by Reading on the one hand and in the BHZ poset introduced by Bergeron, Zabrocki and the third author on the other hand, are the same. We also show that these two partial orders are related by an equality between generating series for their M{\"o}bius numbers, and provide a dimension-preserving bijection between the order complex on the BHZ poset and the pulling triangulation of the permutahedron arising from the right weak order, analogous to the bijection defined by Reading between the order complex of the shard poset and the same triangulation of the permutahedron., Comment: 14 pages, 2 figures

Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let \lambda be a dominant integral weight, let T_\lambda = {t_\lambda} be the crystal with one element of weight \lambda, and let B(\lambda) \subset B(\infty) \otimes T_\lambda be the crystal of the integrable representation of highest weight \lambda. We compute the descending string parameters of an element b \otimes t_\lambda in B(\lambda) in terms of the Lusztig parameters of b., Comment: 4 pages; in French

We give a realization of the infinity crystal for affine sl(2) using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of the infinity crystal in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the analogues in this case of the Mirkovic-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the only other rank two affine Kac-Moody algebra., Comment: 24 pages

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(infinity) of a symmetrizable Kac-Moody algebra can be endowed; these orders are defined so that the transition matrices between bases naturally indexed by B(infinity) are lower triangular., Comment: References updated

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(-\infty), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirkovi\'c-Vilonen polytope, can be associated to each element in B(-\infty). 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 \delta. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(-\infty) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra \Lambda\ of the same type as g. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott's tilting theory for the category \Lambda-mod. The partitions we need come from studying the category of semistable \Lambda-modules of dimension-vector a multiple of \delta., Comment: 104 pages

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.

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver., Comment: 39 pages

Let $G$ be a complex reductive group and let $G^\vee$ be its Langlands dual. Let us choose a triangular decomposition $\mathfrak g^\vee=\mathfrak n^\vee_-\oplus\mathfrak h^\vee\oplus\mathfrak n^\vee_+$ of the Lie algebra $G^\vee$. Braverman, Finkelberg and Gaitsgory show that the set of all Mirkovi\'c-Vilonen cycles in the affine grassmannian $\mathscr G=G\bigl(\mathbb C((t))\bigr)/G\bigl(\mathbb C[[t]]\bigr)$ is a crystal isomorphic to the crystal of the canonical basis of $U(\mathfrak n^\vee_+)$. 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 any MV cycle can be obtained as the closure of one of the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals., Comment: This is the very new version

We propose an analogue of Solomon's descent theory for the case of a wreath product G ~ S_n, 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.

To an abelian category A of homological dimension 1 satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth projective curve defined over a finite field, and observed analogies with quantum affine algebras. We recover here in an elementary way his results in the case when the curve is the projective line., Comment: 29 pages