Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-29T06:12:12.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Exterior Powers of the Adjoint Representation

Published online by Cambridge University Press:  20 November 2018

Mark Reeder
Mark Reeder*
Affiliation:
University of Oklahoma, Dept. of Mathematics, Norman, Oklahoma, USA 73019 e-mail: mreeder@uoknor.edu
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.

Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.

Keywords

20G0520C3022E1022E60Lie algebrasadjoint representationexterior algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 133 - 159
Copyright
Copyright © Canadian Mathematical Society 1997

References

[B] Bourbaki, N., Groupes et Algèbres le Lie, IV, V, VI, Hermann, Paris, 1968.Google Scholar
[BL] Benyon, W., Lusztig, G., Some numerical results on the characters of exceptional Weyl groups, Math. Proc. Cambridge Phil. Soc. 84(1978), 417–426.Google Scholar
[Br1] Broer, B., Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113(1993), 1–20.Google Scholar
[Br2] Broer, B., The sum of generalized exponents and Chevalley's restriction theoremfor modules of covariants, Indag. Math. N.S. 6(1995), 385–396.Google Scholar
[BMP] Bremner, M.R., Moody, R.V., Patera, J., Tables of dominant weight multiplicities for representations of simple Lie algebras, Monographs and Textbooks in Pure and Applied Mathematics 90, Marcel Dekker, Inc., New York, 1985.Google Scholar
[Ch] Chevalley, C., The Betti numbers of the simple Lie groups, Proc. Int. Math. Cong. II(1950), 21–24.Google Scholar
[CE] Chevalley, C., Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. A.M.S. 63(1948), 85–124.Google Scholar
[Co] Coleman, A.J., The Betti numbers of the simple Lie groups, Canad. J. Math. 10(1958), 349–356.Google Scholar
[Gi] Ginzburg, V., Perverse sheaves on a loop group and Langlands’ duality, preprint.Google Scholar
[G1] Gupta, R.K., Characters and the q-analogue of weight multiplicities, J. London Math. Soc. (2) 36(1987), 68–76.Google Scholar
[G2] Gupta, R.K., Generalized exponents via Hall-Littlewood symmetric functions, Bull. A.M.S. 16(1987), 287– 291.Google Scholar
[H] Hesselink, W.H., Characters of the nullcone, Math. Ann. 252(1980), 179–182.Google Scholar
[K] Kempf, G., On the collapsing of homogeneous bundles, Inv. Math. 37(1976), 229–239.Google Scholar
[Ka] Kato, S.I., Spherical functions and a q-analogue of Kostant's weight multiplicity formula, Inv. Math. 66(1982), 461–468.Google Scholar
[Ki] Kirillov, A.A., Polynomial covariants of the symmetric group and some of its analoges, Funk. Anal. Pril. 18(1984), 74–75.Google Scholar
[Ko1] Kostant, B., Lie group representations on polynomial rings, Am. Jn. Math 85(1963), 327–404.Google Scholar
[Ko2] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math 74(1961), 329– 387.Google Scholar
[M] Macdonald, I., Spherical functions on a group of p-adic type, Ramanujan Institute Publications, 1971.Google Scholar
[M2] Macdonald, I., Symmetric functions and Hall polynomials, Oxford Univ. Press, 1979.Google Scholar
[Ma] Matsuzawa, J.I., On the generalized exponents of classical Lie groups, Comm. Alg. (12) 16(1988), 2579–2623.Google Scholar
[N] Najafi, M., Clifford algebra structure on the cohomology algebra of compact symmetric spaces, Master’s thesis, M.I.T., 1979.Google Scholar
[PRV] Parasarathy, K.R., Ranga-Rao, R.,Varadarajan, V.S., Representations of complex semisimple Lie groups and Lie algebras, Ann. Math. 85(1967), 383–429.Google Scholar
[R1] Reeder, M., The cohomology of compact Lie groups, L’Ens. Math. 41(1995), 181–200.Google Scholar
[R2] Reeder, M., p-adicWhittaker functions and vector bundles on flag manifolds, Compositio Math. (1) 85(1993), 9–36.Google Scholar
[R3] Reeder, M., Whittaker functions, prehomogeneous vector spaces and standard representations of p-adic groups, J. Reine Angew. Math 450(1994), 83–121.Google Scholar
[So] Solomon, L., Invariants of finite reflection groups, Nagoya Jn. Math. 22(1963), 57–64.Google Scholar
[St] Stembridge, J., First layer formulas for characters of SL(n, ℂ), T.A.M.S. 299(1987), 319–350.Google Scholar
[V] Varadarajan, V.S., Lie groups, Lie algebras and their representations, Springer-Verlag, 1984.Google Scholar