Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T22:56:18.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Algorithms for representation theory of real reductive groups

Part of: Algebraic combinatorics Lie groups

Published online by Cambridge University Press:  06 January 2009

Jeffrey Adams  and
Fokko du Cloux
Jeffrey Adams
Affiliation:
Department of Mathematics, Mathematics Building, University of Maryland, College Park, MD 20742-4015, USA (jda@math.umd.edu)
Fokko du Cloux
Affiliation:
Department of Mathematics, Mathematics Building, University of Maryland, College Park, MD 20742-4015, USA (jda@math.umd.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

Keywords

Lie groupsrepresentation theoryreal reductive groupsalgorithmsadmissible representationscomputational representation theory

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields 05E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 2 , April 2009 , pp. 209 - 259
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adams, J., Lifting of characters, in Harmonic Analysis on Reductive Groups, Brunswick, ME, 1989, Progress in Mathematics, Volume 101, pp. 1–50 (Birkhäuser, Boston, MA, 1991).CrossRefGoogle Scholar
2.Adams, J., Guide to the Atlas software: computational representation theory of real reductive groups, in Proc. Conf. Representation Theory of Real Reductive Groups, Snowbird, 2006, Contemporary Mathematics (American Mathematical Society, Providence, RI, 2008).Google Scholar
3.Adams, J. and Vogan, D. A. Jr, L-groups, projective representations, and the Langlands classification, Am. J. Math. 114(1) (1992), 45–138.CrossRefGoogle Scholar
4.Adams, J., Barbasch, D. and Vogan, D. A. Jr, The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, Volume 104 (Birkhäuser, Boston, MA, 1992).CrossRefGoogle Scholar
5.Beĭlinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers, in Analysis and Topology on Singular Spaces, I, Luminy, 1981, Astérisque, Volume 100, pp. 5–171 (Société Mathématique de France, Paris, 1982).Google Scholar
6.Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, in Automorphic Forms, Representations and L-Functions, Corvallis, OR, 1977, Part 1 (with a supplement, ‘On the notion of an automorphic representation’, by R. P. Langlands), Proceedings of Symposia in Pure Mathematics, Volume 33, pp. 189–207 (American Mathematical Society, Providence, RI, 1979).Google Scholar
7.Bourbaki, N., Groupes et algèbres de Lie, in Éléments de mathématique, Chapters 4–6 (Masson, New York, 1981).Google Scholar
8.Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields, Annals Math. (2) 103(1) (1976), 103–161.CrossRefGoogle Scholar
9.du Cloux, F., Combinatorics for the representation theory of reductive groups, preprint (available at www.liegroups.org/papers/combinatorics.pdf, 2006).Google Scholar
10.Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, Volume 34 (American Mathematical Society, Providence, RI, 2001; corrected reprint of the 1978 original).Google Scholar
11.Humphreys, J. E., Linear algebraic groups, Graduate Texts in Mathematics, Volume 21 (Springer, 1975).CrossRefGoogle Scholar
12.Knapp, A. W., Wey1 group of a cuspidal parabolic, Annales Scient. Éc. Norm. Sup. 8(2) (1975), 275–294.CrossRefGoogle Scholar
13.Knapp, A. W., Representation theory of semisimple groups: an overview based on examples (Princeton University Press, 1986).Google Scholar
14.Kostant, B., On the conjugacy of real Cartan subalgebras, I, Proc. Natl Acad. Sci. USA 41 (1955), 967–970.Google Scholar
15.Kostant, B., On the conjugacy of real Cartan subalgebras, II, preprint (1955, to appear in Kostant's collected works).CrossRefGoogle Scholar
16.Langlands, R. P., On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical Surveys and Monographs, Volume 31, pp. 101–170 (American Mathematical Society, Providence, RI, 1989).CrossRefGoogle Scholar
17.Matsuki, T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Jpn 31(2) (1979), 331–357.CrossRefGoogle Scholar
18.Miličić, D., Algebraic -modules and representation theory of semisimple Lie groups, in The Penrose Transform and Analytic Cohomology in Representation Theory, South Hadley, MA, 1992, Contemporary Mathematics, Volume 154, pp. 133–168 (American Mathematical Society, Providence, RI, 1993).Google Scholar
19.Onishchik, A. L. and Vinberg, È. B., Lie groups and algebraic groups, Springer Series in Soviet Mathematics (Springer, 1990; translated from the Russian and with a preface by Leites, D. A.).CrossRefGoogle Scholar
20.Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, Volume 139 (Academic Press, 1994; translated from the 1991 Russian original by Rowen, R.).CrossRefGoogle Scholar
21.Richardson, R. W. and Springer, T. A., The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), 389–436.CrossRefGoogle Scholar
22.Schmid, W., On the characters of the discrete series, Invent. Math. 30 (1975), 47–144.CrossRefGoogle Scholar
23.Springer, T. A., Linear algebraic groups, 2nd edn, Progress in Mathematics, Volume 9 (Birkhäuser, Boston, MA, 1998).CrossRefGoogle Scholar
24.Sugiura, M., Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Jpn 11 (1959), 374–434.CrossRefGoogle Scholar
25.Tits, J., Normalisateurs de tores, I, Groupes de Coxeter étendus, J. Alg. 4 (1966), 96–116.CrossRefGoogle Scholar
26.Vogan, D. A. Jr, Representations of Real Reductive Lie Groups, Progress in Mathematics, Volume 15 (Birkhäuser, Boston, MA, 1981).Google Scholar
27.Vogan, D. A. Jr, Irreducible characters of semisimple Lie groups, IV, Character-multiplicity duality, Duke Math. J. 49(4) (1982), 943–1073.CrossRefGoogle Scholar
28.Zuckerman, G., Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Annals Math. (2) 106(2) (1977), 295–308.Google Scholar