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.


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
Full Text: DOI arXiv Link


[1] J. E. Anderson, ”On Mirković and Vilonen’s Intersection Homology cycles for the Loop Grassmannian,” PhD Thesis , Princeton University, 2000.
[2] J. E. Anderson, ”A polytope calculus for semisimple groups,” Duke Math. J., vol. 116, iss. 3, pp. 567-588, 2003. · Zbl 1064.20047
[3] J. Anderson and M. Kogan, ”Mirković-Vilonen cycles and polytopes in Type A,” Internat. Math. Res. Not., iss. 12, pp. 561-591, 2004. · Zbl 1082.14012
[4] J. Anderson and M. Kogan, ”The algebra of Mirković-Vilonen cycles in type A,” Pure Appl. Math. Q., vol. 2, iss. 4, part 2, pp. 1187-1215, 2006. · Zbl 1110.14043
[5] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves. · Zbl 0864.14007
[6] A. Berenstein, S. Fomin, and A. Zelevinsky, ”Parametrizations of canonical bases and totally positive matrices,” Adv. Math., vol. 122, iss. 1, pp. 49-149, 1996. · Zbl 0966.17011
[7] A. Berenstein, S. Fomin, and A. Zelevinsky, ”Cluster algebras. III. Upper bounds and double Bruhat cells,” Duke Math. J., vol. 126, iss. 1, pp. 1-52, 2005. · Zbl 1135.16013
[8] A. Berenstein and A. Zelevinsky, ”Total positivity in Schubert varieties,” Comment. Math. Helv., vol. 72, iss. 1, pp. 128-166, 1997. · Zbl 0891.20030
[9] A. Berenstein and A. Zelevinsky, ”Tensor product multiplicities, canonical and totally positive varieties,” Invent. Math., vol. 143, iss. 1, pp. 77-128, 2001. · Zbl 1061.17006
[10] A. Braverman and D. Gaitsgory, ”Crystals via the affine Grassmannian,” Duke Math. J., vol. 107, iss. 3, pp. 561-575, 2001. · Zbl 1015.20030
[11] A. Braverman, M. Finkelberg, and D. Gaitsgory, ”Uhlenbeck spaces via affine Lie algebras,” in The Unity of Mathematics, Boston, MA: Birkhäuser, 2006, pp. 17-135. · Zbl 1105.14013
[12] G. Ewald, Combinatorial Convexity and Algebraic Geometry, New York: Springer-Verlag, 1996. · Zbl 0869.52001
[13] M. Finkelberg and I. Mirković, ”Semi-infinite flags. I. Case of global curve \(\mathbb P^1\),” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Providence, RI: Amer. Math. Soc., 1999, pp. 81-112. · Zbl 1076.14512
[14] E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Second ed., Providence, RI: Amer. Math. Soc., 2004. · Zbl 1106.17035
[15] S. Fomin and A. Zelevinsky, ”Double Bruhat cells and total positivity,” J. Amer. Math. Soc., vol. 12, iss. 2, pp. 335-380, 1999. · Zbl 0913.22011
[16] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993. · Zbl 0813.14039
[17] S. Gaussent and P. Littelmann, ”LS galleries, the path model, and MV cycles,” Duke Math. J., vol. 127, iss. 1, pp. 35-88, 2005. · Zbl 1078.22007
[18] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality. · Zbl 0736.22009
[19] J. Kamnitzer, ”The crystal structure on the set of Mirković-Vilonen polytopes,” Adv. Math., vol. 215, iss. 1, pp. 66-93, 2007. · Zbl 1134.14028
[20] G. Lusztig, ”Singularities, character formulas, and a \(q\)-analog of weight multiplicities,” in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Paris: Soc. Math. France, 1983, pp. 208-229. · Zbl 0561.22013
[21] G. Lusztig, ”Canonical bases arising from quantized enveloping algebras,” J. Amer. Math. Soc., vol. 3, iss. 2, pp. 447-498, 1990. · Zbl 0703.17008
[22] G. Lusztig, ”Introduction to quantized enveloping algebras,” in New Developments in Lie Theory and their Applications (Córdoba, 1989), Boston, MA: Birkhäuser, 1992, pp. 49-65. · Zbl 0767.17014
[23] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993. · Zbl 0788.17010
[24] G. Lusztig, ”An algebraic-geometric parametrization of the canonical basis,” Adv. Math., vol. 120, iss. 1, pp. 173-190, 1996. · Zbl 0877.17006
[25] I. Mirković and K. Vilonen, ”Perverse sheaves on affine Grassmannians and Langlands duality,” Math. Res. Lett., vol. 7, iss. 1, pp. 13-24, 2000. · Zbl 0987.14015
[26] I. Mirković and K. Vilonen, ”Geometric Langlands duality and representations of algebraic groups over commutative rings,” Ann. of Math., vol. 166, iss. 1, pp. 95-143, 2007. · Zbl 1138.22013
[27] D. E. Speyer, ”Horn’s problem, Vinnikov curves, and the hive cone,” Duke Math. J., vol. 127, iss. 3, pp. 395-427, 2005. · Zbl 1069.14037
[28] D. Speyer and B. Sturmfels, ”The tropical Grassmannian,” Adv. Geom., vol. 4, iss. 3, pp. 389-411, 2004. · Zbl 1065.14071
[29] G. M. Ziegler, Lectures on Polytopes, New York: Springer-Verlag, 1995. · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.