Duke Mathematical Journal

LS galleries, the path model, and MV cycles

S. Gaussent and P. Littelmann

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give an interpretation of the path model of a representation (see [18]) of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by Lakshmibai-Seshadri (LS) galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure-Hansen-Bott-Samelson desingularization $\hat\Sigma(\lambda)$ of the closure of an orbit $G(\mathbb{C}[[t]])\cdot\lambda$ in the affine Grassmannian. The homology of $\hat\Sigma(\lambda)$ has a basis given by Białynicki-Birula (BB) cells, which are indexed by the T-fixed points in $\hat\Sigma(\lambda)$. Now the points of $\hat\Sigma(\lambda)$ can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T-fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a nonempty intersection with $G(\mathbb{C}[[t]])\cdot\lambda$ (identified with an open subset of $\hat\Sigma(\lambda)$), and we show that the closures of the strata associated to LS galleries are exactly the Mirković-Vilonen (MV) cycles (see [25]) which form a basis of the representation V(λ) for the Langlands dual group $G^\vee$.

Article information

Source
Duke Math. J., Volume 127, Number 1 (2005), 35-88.

Dates
First available in Project Euclid: 4 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1109963910

Digital Object Identifier
doi:10.1215/S0012-7094-04-12712-5

Mathematical Reviews number (MathSciNet)
MR2126496

Zentralblatt MATH identifier
1078.22007

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

Citation

Gaussent, S.; Littelmann, P. LS galleries, the path model, and MV cycles. Duke Math. J. 127 (2005), no. 1, 35--88. doi:10.1215/S0012-7094-04-12712-5. https://projecteuclid.org/euclid.dmj/1109963910


Export citation

References

  • \lccJ. E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), 567–588.
  • \lccJ. Anderson and M. Kogan, Mirković-Vilonen cycles and polytopes in type A, Int. Math. Res. Not. 2004, no. 12, 561–591.
  • \lccA. Beauville and Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385–419.
  • \lccI. N. Bernšteĭn, I. M. Gelfand, and S. I. Gelfand, Structure of representations that are generated by vectors of higher weight, Funct. Anal. Appl. 5 (1971), 1–9.
  • \lccA. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.
  • ––––, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 667–674.
  • \lccA. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), 561–575.
  • \lccK. S. Brown, Buildings, Springer, New York, 1989.
  • \lccF. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251.
  • \lccC. Contou-Carrère, Géométrie des groupes semi-simples, résolutions équivariantes et lieu singulier de leurs variétés de Schubert, Thèse d'état, Université Montpellier II, 1983.
  • \lccR. Dabrowski, “A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety” in Kazhdan-Lusztig Theory and Related Topics (Chicago, 1989), Contemp. Math. 139, Amer. Math. Soc., Providence, 1992, 113–120.
  • \lccV. V. Deodhar, On some geometric aspects of Bruhat orderings, I: A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), 499–511.
  • \lccN. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $\mathfrak{p}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. 25 (1965), 5–48.
  • \lccA. Joseph, Quantum Groups and Their Primitive Ideals, Ergeb. Math. Grenzgeb. (3) 29, Springer, Berlin, 1995.
  • \lccM. Kashiwara, “Similarities of crystal bases” in Lie Algebras and Their Representations (Seoul, 1995), Contemp. Math. 194, Amer. Math. Soc., Providence, 1996, 177–186.
  • \lccS. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math. 204, Birkhäuser, Boston, 2002.
  • \lccY. Laszlo and C. Sorger, The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. École Norm. Sup. 30 (1997), 499–525.
  • \lccP. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), 329–346.
  • ––––, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499–525.
  • ––––, “Characters of representations and paths in ${H}_{\mathbb{R}}^*$” in Representation Theory and Automorphic Forms (Edinburgh, 1996), Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, 1997, 29–49.
  • ––––, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), 551–567.
  • ––––, “Bases for representations, LS-paths and Verma flags” in A Tribute to C. S. Seshadri's (Chennai, India, 2002), Trends Math., Birkhäuser, Basel, 2003, 323–345.
  • \lccG. Lusztig, “Singularities, character formulas, and a $q$-analog of weight multiplicities” in Analysis and Topology on Singular Spaces, II, III (Luminy, France, 1981), Astérisque 101102, Soc. Math. France, Montrouge, 1983, 208–229.
  • ––––, An algebraic-geometric parametrization of the canonical basis, Adv. Math. 120 (1996), 173–190.
  • \lccI. Mirković and K. Vilonen, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13–24.
  • \lccB. C. Ngô and P. Polo, Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Algebraic Geom. 10 (2001), 515–547.
  • \lccM. Ronan, Lectures on Buildings, Perspect. Math. 7, Academic Press, Boston, 1989.
  • \lccJ. Tits, Buildings of Spherical Type and Finite $BN$-Pairs, Lecture Notes in Math. 386, Springer, Berlin, 1974.
  • ––––, Résumé de cours, Ann. du Collège France 82 (1981/82), 91–105.
  • ––––, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542–573.