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 Mirković-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.
We show that generalized orbital varieties for Mirkovic-Vybornov slices can be indexed by semi-standard Young tableaux. We also check that the Mirkovic-Vybornov isomorphism sends generalized orbital varieties to (dense subsets of) Mirkovic-Vilonen cycles, such that the (combinatorial) Lusztig datum of a generalized orbital variety, which it inherits from its tableau, is equal to the (geometric) Lusztig datum of its MV cycle.
Let G be a reductive connected algebraic group over the field of complex numbers. Through the geometric Satake equivalence, the fundamental classes of the Mirković-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.
We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirković-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirković-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of $\mathfrak{sl}_n$ representations.
We prove the Mirković-Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau-Mautner-Williamson theory of parity sheaves.
We give an explicit description of the (lowering) Kashiwara operators on Mirković-Vilonen polytopes in types $B$ and $C$, which provides a simple method for generating Mirković-Vilonen polytopes inductively. This description can be thought of as a modification of the original Anderson-Mirković conjecture, which Kamnitzer proved in the case of type $A$, and presented a counterexample in the case of type $C_{3}$.
In the current paper, we give a quiver theoretical interpretation of Mirković-Vilonen polytopes in type $A_n$. As a by-product, we give a new proof of the Anderson-Mirković conjecture which describes the explicit forms of the actions of lowering Kashiwara operators on the set of Mirković-Vilonen polytopes.
Formulas for the product of an irreducible character $χ_λ$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal $\mathcal{B}(λ+ρ)$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of $p$-adic groups as a sum over the canonical basis $\mathcal{B}(-\infty)$, which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the correspondi…
In this paper, we give a polytopal estimate of Mirković-Vilonen polytopes lying in a Demazure crystal in terms of Minkowski sums of extremal Mirković-Vilonen polytopes. As an immediate consequence of this result, we provide a necessary (but not sufficient) polytopal condition for a Mirković-Vilonen polytope to lie in a Demazure crystal.
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes.
Let Gr be the affine Grassmannian for a connected complex reductive group G. Let C_G be the complex vector space spanned by (equivalence classes of) Mirkovic-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a convolution product on MV-cycles, making C_G into a commutative algebra. We show, in type A, that C_G isomorphic to C[N], the algebra of functions on the unipotent radical N of a Borel subgroup of G; then each MV-cycle defines a polynomial in C[N], which we call an MV-polynomial. We conjecture that those MV-polynomials which are cluster monomials for a Fomin…
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…
Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $σ$ be a Dynkin automorphism on $G$. In this paper, we get a bijection between the set of $\st$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\st$, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula.
We study the description of the crystal structure on the set of Mirković-Vilonen polytopes. Anderson and Mirković defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for type A and gave an counterexample for type C_{3}. He also gave an explicit formula to calculate the Kashiwara operator for type A. In this paper we prove that a part of the AM conjecture still holds in general, answering an open question of Kamnitzer (2007). Moreover, we show that although the formula given by Kamnitzer does not hold in general, it is still valid i…
We describe how Mirkovic-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of the KLR algebra and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense for finite dimensional semi-simple Lie algebras, but our construction actually gives a map from the infinity crystal to polytopes in all symmetrizable Kac-Moody algebras. However, to make the map injective and have well-defined crystal…
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 Mirković-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…
We study a class of varieties which generalize the classical orbital varieties of Joseph. We show that our generalized orbital varieties are the irreducible components of a Mirkovic-Vybornov slice to a nilpotent orbit, and can be labeled by semistandard Young tableaux. Furthermore, we prove that Mirkovic-Vilonen cycles are obtained by applying the Mirkovic-Vybornov isomorphism to generalized orbital varieties and taking a projective closure, refining Mirkovic and Vybornov's result. As a consequence, we are able to use the Lusztig datum of a Mirkovic--Vilonen cycle to determine the tableau labe…
We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central degeneration of semi-infinite orbits and explain its relations with Levi restriction. Also, we discuss the central degeneration of Mirkovi$\acute{\text{c}}$-Vilonen cycles in the affine Grassmannian, and the corresponding transformations of Mirkovi$\acute{\text{c}}$-Vilonen polytopes. In addition, we shed some light on the geometry of Iwahori MV cycles in the aff…
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they may be used to compute weight multiplicities and tensor product multiplicities for representations of a semisimple group. The polytopes are explicitly described for a few low rank groups.
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…
