## 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
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### References:

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