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Representation Theory

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On Mirković-Vilonen cycles and crystal combinatorics

Authors: Pierre Baumann and Stéphane Gaussent
Journal: Represent. Theory 12 (2008), 83-130
MSC (2000): Primary 20G05; Secondary 05E15, 14M15, 17B10, 22E67
Published electronically: March 5, 2008
MathSciNet review: 2390669
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Abstract: 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.

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Additional Information

Pierre Baumann
Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Stéphane Gaussent
Affiliation: Institut Élie Cartan, Unité Mixte de Recherche 7502, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France

Keywords: Affine Grassmannian, Mirković-Vilonen cycle, crystal
Received by editor(s): April 26, 2007
Received by editor(s) in revised form: October 17, 2007
Published electronically: March 5, 2008
Additional Notes: Both authors are members of the European Research Training Network “LieGrits”, contract no. MRTN-CT 2003-505078.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.