On Mirkovi\'c-Vilonen cycles and crystals combinatorics

TL;DR

This paper explores the combinatorics of Mirković-Vilonen cycles and their relation to crystals, providing explicit descriptions and establishing isomorphisms with canonical bases and LS paths.

Contribution

It offers an explicit description of MV cycles from string parameters and proves the crystal isomorphism between LS paths and MV cycles.

Findings

Explicit description of dense MV cycle subsets from string parameters

Any MV cycle can be obtained from Lusztig's parametrization

The LS path and MV cycle bijection is a crystal isomorphism

Abstract

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.