2006/06/28 by Pierre Baumann, Baumann, Pierre, Stéphane Gaussent +1 · 1 citation
Mathematics · #20G05 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0606711
openalex publication_date 2006/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-⊕\mathfrak h^\vee⊕\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(\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.