On Mirković-Vilonen cycles and crystal combinatorics

Let $G$ be a complex connected reductive group and let $G^{\vee}$ be its Langlands dual. Let us choose a triangular decomposition ${\mathfrak{n}}^{-,{\vee}} \oplus{\mathfrak{h}}^{\vee}\oplus{\mathfrak{n}}^{+,{\vee}}$ of the Lie algebra of $G^{\vee}$. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannian $G\bigl(\mathbb{C}((t))\bigr)/G\bigl(\mathbb{C}[[t]]\bigr)$ 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 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.