Abstract
Let G be a reductive algebraic group over an algebraically closed field \( \kappa\) of prime characteristic p. The first question that presents itself when we look at finite-dimensional representations of G is the problem of how to determine the irreducible characters. This is the problem we want to discuss in this lecture.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
H. H. Andersen,The strong linkage principle, J. Reine Angew. Math.,315 (1980), 53–59.
H. H. Andersen,The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B, Ann. of Math. (2),112 (1980), 113–121.
H. H. Andersen, J. C. Jantzen, and W. Soergel,Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: Independence of p, Astérisque,220 (1994).
H. H. Andersen, P. Polo, and Wen K.,Representations of quantum algebras, Invent. Math.,104 (1991), 1–59.
AH. H. Andersen and Wen K.,Representations of quantum algebras. The mixed case, J. Reine Angew. Math.,427 (1992), 35–50.
A. Beilinson and J. Bernstein,Localisation de\( g\)-modules, C. R. Acad. Sci. Paris,292 (1981), 15–18.
J.-L. Brylinski and M. Kashiwara,Kazhdan-Lusztig conjecture and holonomic system, Invent. Math.,64 (1981), 387–410.
L. Casian,Formule de multiplicité de Kazhdan-Lusztig dans le case de Kac-Moody, C.R. Acad. Sci. Paris,310 (1990), 333–337.
L. Casian,Kazhdan-Lusztig conjecture in the negative level case (Kac-Moody algebras of affine type), preprint.
C. Chevalley,Les systèmes linéaires sur G/B, Exp. 15 in: Sem. Chevalley ( Classification des groupes de Lie algébriques ), (1957), Paris.
C. W. Curtis,Representations of Lie algebras of classical type with application to linear groups, J. Math. Mech.,9 (1960), 307–326.
V. Deodhar, O. Gabber, and V. Kac,Structure of some categories of infinite- dimensional Lie algebras, Adv. in Math.,45 (1982), 92–116.
V. G. Drinfeld,On quasitriangular quasi-Hopf algebras closely related to Gal(Q/Q), Algebra i Analiz 2, (1990), 149–181.
W. Haboush,A short proof of the Kempf vanishing theorem, Invent. Math.,56 (1980), 109–112.
J. E. Humphreys, Ordinary and modular representations of Chevalley groups,Lecture Notes in Math.,528 (1976), Springer, Berlin and New York.
J. C. Jantzen, Representations of algebraic groups, Pure and Appl. Math.,131, (1987) Academic Press, Boston MA.
V. Kac, Infinite Dimensional Lie Algebras, Third edition, (1990), Cambridge Univ. Press, London and New York.
M. Kashiwara,Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra, in: The Grothendieck Festschrift, Vol II, Progr. Math.,87, (1990) Birkhäuser, Basel and Boston, 407–433.
M. Kashiwara,Crystal base and Littelmann’s refined Demazure character formula, Duke Math. J.,71 (1993), 839–858.
M. Kashiwara and T. Tanisaki,Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra II, Progr. Math.,92, (1990) Birkhäuser, Basel and Boston, 159–195.
M. Kashiwara and T. Tanisaki,Characters of the negative level highest weight modules for affine Lie algebras, preprint (1994).
S. Kato,On the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math.,55 (1985), 103–130.
D. Kazhdan and G. Lusztig,Representations of Coxeter groups and Hecke algebras, Invent. Math.,53 (1979), 165–184.
D. Kazhdan and G. Lusztig,Schubert varieties and Poincaré duality, Proc. Sym- pos. Pure Math.,36 (1980), 185–203.
D. Kazhdan and G. Lusztig,Tensor structures arising from affine Lie algebras I-IV, J. Amer. Math. Soc.,6 (1994), 905–947.
D. Kazhdan and G. Lusztig,Tensor structures arising from affine Lie algebras I-IV, J. Amer. Math. Soc.,6 (1994), 949–1011.
D. Kazhdan and G. Lusztig,Tensor structures arising from affine Lie algebras I-IV, J. Amer. Math. Soc.,7 (1994), 335–381.
D. Kazhdan and G. Lusztig,Tensor structures arising from affine Lie algebras I-IV, J. Amer. Math. Soc.,7 (1994), 383–453.
G. Kempf,Linear systems on homogeneous spaces, Ann. of Math (2),103 (1976), 557–591.
S. Kumar,Toward proof of Lusztig’s conjecture concerning negative level representations of affine Lie algebras, J. Algebra,164 (1994), 515–527.
G. Lusztig,Some problems in the representation theory of finite Chevalley groups, Proc. Sympos. Pure Math.,37 (1980), 313–318.
G. Lusztig,Modular representations and quantum groups, Contemp. Math.,82 (1989), 58–77.
G. Lusztig,On quantum groups, J. Algebra 131 (1990), 466–475.
G. Lusztig, Introduction to quantum groups, Progr. Math.,110 (1993), Birkhäuser, Basel and Boston.
G. Lusztig,Monodromic systems on affine flag manifolds, Proc. Roy. Soc. London series A (1994),445, 231–246.
B. Parshall and Wang J.-P.,Quantum linear groups, Mem. Amer. Math. Soc.,439 (1991).
S. Ryom-Hansen,A q-analogue of Kempf’s vanishing theorem, preprint (1994).
W. Soergel,Kategorie O, perverse Garben und Modulen über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc.,3 (1990), 421–445.
R. Steinberg,Representations of algebraic groups, Nagoya Math. J.,22 (1963), 33–56.
L. Thams,Two classical results in the quantum mixed case, J. Reine Angew. Math.,436 (1993), 129–153.
D. Vogan, Representations of real reductive Lie groups, Progr. Math.,15 (1981), Birkhäuser, Basel and Boston.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhäser Verlag, Basel, Switzerland
About this paper
Cite this paper
Andersen, H.H. (1995). The Irreducible Characters for Semi-Simple Algebraic Groups and for Quantum Groups. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_66
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9078-6_66
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9897-3
Online ISBN: 978-3-0348-9078-6
eBook Packages: Springer Book Archive