On Mirković-Vilonen cycles and crystal combinatorics
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- by Pierre Baumann and Stéphane Gaussent
- Represent. Theory 12 (2008), 83-130
- DOI: https://doi.org/10.1090/S1088-4165-08-00322-1
- Published electronically: March 5, 2008
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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.References
- Jared E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3, 567–588. MR 1958098, DOI 10.1215/S0012-7094-03-11636-1
- Arnaud Beauville and Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419. MR 1289330, DOI 10.1007/BF02101707
- A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint available at http\string://www.math.uchicago.edu/\string mitya/langlands.html.
- Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49–149. MR 1405449, DOI 10.1006/aima.1996.0057
- Arkady Berenstein and Andrei Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128–166. MR 1456321, DOI 10.1007/PL00000363
- Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
- A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
- Alexander Braverman, Michael Finkelberg, and Dennis Gaitsgory, Uhlenbeck spaces via affine Lie algebras, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 17–135. MR 2181803, DOI 10.1007/0-8176-4467-9_{2}
- Alexander Braverman and Dennis Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), no. 3, 561–575. MR 1828302, DOI 10.1215/S0012-7094-01-10736-9
- F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923, DOI 10.1007/BF02715544
- S. Gaussent and P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (2005), no. 1, 35–88. MR 2126496, DOI 10.1215/S0012-7094-04-12712-5
- V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, preprint arXiv:alg-geom/9511007.
- J. Kamnitzer, Mirković-Vilonen cycles and polytopes, preprint arXiv:math.AG/0501365.
- Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485. MR 1203234, DOI 10.1215/S0012-7094-93-06920-7
- Masaki Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858. MR 1240605, DOI 10.1215/S0012-7094-93-07131-1
- Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
- Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9–36. MR 1458969, DOI 10.1215/S0012-7094-97-08902-X
- Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- George Lusztig, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- George Lusztig, Introduction to quantized enveloping algebras, New developments in Lie theory and their applications (Córdoba, 1989) Progr. Math., vol. 105, Birkhäuser Boston, Boston, MA, 1992, pp. 49–65. MR 1190735, DOI 10.1007/978-1-4612-2978-0_{3}
- George Lusztig, An algebraic-geometric parametrization of the canonical basis, Adv. Math. 120 (1996), no. 1, 173–190. MR 1392278, DOI 10.1006/aima.1996.0036
- Ivan Mirković and Kari Vilonen, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), no. 1, 13–24. MR 1748284, DOI 10.4310/MRL.2000.v7.n1.a2
- I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. MR 2342692, DOI 10.4007/annals.2007.166.95
- Sophie Morier-Genoud, Relèvement géométrique de la base canonique et involution de Schützenberger, C. R. Math. Acad. Sci. Paris 337 (2003), no. 6, 371–374 (French, with English and French summaries). MR 2015078, DOI 10.1016/j.crma.2003.07.001
- Yoshihisa Saito, PBW basis of quantized universal enveloping algebras, Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, 209–232. MR 1265471, DOI 10.2977/prims/1195166130
- Mitsuhiro Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981), no. 8, 841–882. MR 611561, DOI 10.1080/00927878108822621
- Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
Bibliographic Information
- Pierre Baumann
- Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- Email: baumann@math.u-strasbg.fr
- 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
- Email: Stephane.Gaussent@iecn.u-nancy.fr
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 12 (2008), 83-130
- MSC (2000): Primary 20G05; Secondary 05E15, 14M15, 17B10, 22E67
- DOI: https://doi.org/10.1090/S1088-4165-08-00322-1
- MathSciNet review: 2390669