Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T09:28:35.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Green Functions and Unipotent Classes for Finite Reductive Groups, I

Published online by Cambridge University Press:  11 January 2016

David Hernandez  and
Hiraku Nakajima
David Hernandez
Affiliation:
CNRS - UMR 8100: Laboratoire de Mathématiques de Versailles45 avenue des Etats-Unis, Bat. Fermat, 78035 VersaillesFrancehernandez@math.cnrs.fr
Hiraku Nakajima
Affiliation:
Department of MathematicsKyoto UniversityKyoto, 606-8502Japannakajima@math.kyoto-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

Keywords

17B3717B65
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 184 , 2006 , pp. 155 - 198
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[B] Bonnafée, C., Sur les caractères des groupes réeductifs finis à centre non connex: applications aux groupes spéeciaux linéeaires et unitaires, preprint.Google Scholar
[BR] Bardsley, P. and Richardson, R. W., Etale slices for algebraic transformation groups in characteristic p, Proc. London Math. Soc., 51 (1985), 295–317.CrossRefGoogle Scholar
[BS] Beynon, W. M. and Spaltenstein, N., Green functions of finite Chevalley groups of type En (n = 6, 7,8), J. Algebra, 88 (1984), 584–614.CrossRefGoogle Scholar
[C] Carter, R., Finite groups of Lie type, Conjugacy classes and complex characters, Wiley-Interscience, New York, 1985.Google Scholar
[L1] Lusztig, G., Characters of Reductive groups over a finite field, Ann. of Math. Studies, Vol. 107, Princeton Univ. Press, Princeton, 1984.Google Scholar
[L2] Lusztig, G., Intersection cohomology complexes on a reductive group, Invent. Math., 75 (1984), 205–272.CrossRefGoogle Scholar
[L3] Lusztig, G., Character sheaves, I, Adv. in Math., 56 (1985), 193–237, II, Adv. in Math., 57 (1985), 226–265, III, Adv. in Math., 57 (1985), 266–315, IV, Adv. in Math., 59 (1986), 1–63, V, Adv. in Math., 61 (1986), 103–155.CrossRefGoogle Scholar
[L4] Lusztig, G., Fourier transforms on a semisimple Lie algebra over Fq , Algebraic groups Utrecht 1986, Lecture Note in Math. 1271, Springer-Verlag (1987), pp. 177–188.Google Scholar
[L5] Lusztig, G., On the representations of reductive groups with disconnected centre, Astéerisque, 168 (1988), 157–168.Google Scholar
[L6] Lusztig, G., A unipotent support for irreducible representations, Adv. in Math., 94 (1992), 139–179.CrossRefGoogle Scholar
[L7] Lusztig, G., Cuspidal local systems and graded Hecke algebras, I, Publ. Math. I.H.E.S., 67 (1988), 145–202.CrossRefGoogle Scholar
[L8] Lusztig, G., Cuspidal local systems and graded Hecke algebras, II, Representations of groups (B. Alliso and G. Cliff, eds.), Canad. Math. Soc. Conf. Proc., Vol. 16, Amer. Math. Soc. (1995), pp. 217–275.Google Scholar
[L9] Lusztig, G., Cuspidal local systems and graded Hecke algebras, III, Representation Theory, 6 (2002), 202–242 (electronic).CrossRefGoogle Scholar
[LS] Lusztig, G. and Spaltenstein, N., On the generalized Springer correspondence for classical groups, Advanced Studies in Pure Math. Vol. 6 (1985), pp. 289–316.CrossRefGoogle Scholar
[M] Malle, G., Green functions for groups of types E6 and F4 in characteristic 2, Comm. in Algebra, 21 (1993), 747–798.Google Scholar
[S1] Shoji, T., On the Green polynomials of Chevalley groups of type F4 , Comm. in Alg., 10 (1982), 505–543.CrossRefGoogle Scholar
[S2] Shoji, T., On the Green polynomials of classical groups, Invent. Math., 74 (1983), 237–267.CrossRefGoogle Scholar
[S3] Shoji, T., Character sheaves and almost characters of reductive groups, Adv. in Math., 111 (1995), 244–313, II, Adv. in Math., 111 (1995), 314–354.Google Scholar
[S4] Shoji, T., Lusztig’s conjecture for finite special linear groups, Representation theory, 10 (2006), 164–222.CrossRefGoogle Scholar
[Spr] Springer, T. A., Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36 (1976), 173–207.CrossRefGoogle Scholar
[W] Waldspurger, J.-L., Une conjecture de Lusztig pour les groupes classiques, Méemoires de la Sociéetée Mathéematique de France 96, SMF, 2004.Google Scholar