On Mirković-Vilonen cycles and crystals combinatorics

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 Mirković-Vilonen cycles in the affine grassmannian $\mathcal{G}=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ 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.

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