Mirković-Vilonen cycles and polytopes. (English) Zbl 1271.20058
Summary: We give an explicit description of the Mirković-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig’s canonical basis and the set of MV polytopes.
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G10 | Cohomology theory for linear algebraic groups |
| 05E10 | Combinatorial aspects of representation theory |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 22E46 | Semisimple Lie groups and their representations |
Keywords:
Mirković-Vilonen cycles; complex reductive groups; intersection homology; affine Grassmannians; Mirković-Vilonen polytopes; Langlands dual; irreducible representations; moment polytopes; Weyl polytopes; Lusztig canonical basesReferences:
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