Abstract
Let ρ: G → GL(V) be a representation of a reductive algebraic group G defined over ℂ. A simple example for such a situation is the natural action of SL 2(ℂ) by coordinate substitution on the vector space R n := ℂ[x,y] n of binary forms of degree n. The G-invariant polynomial functions on V are an important tool to study the orbit structure of the group in the representation space. It was one of the highlights in invariant theory before Hilbert when Gordan proved in 1868 that the ring of invariants is finitely generated in the example considered above. But even for this “simple-looking” example, a complete description of the invariants and the orbits exists only for the cases n ≤ 6 and n = 8.
Supported by Schweizerischer Nationalfonds.
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.
References
Bala, P., Carter, R. W., Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976), 401–425.
Bala, P., Carter, R. W., Classes of unipotent elements in simple algebraic groups II, Math. Proc. Camb. Phil. Soc. 80 (1976). 1–18.
N. Bourbaki, “Groupes et algébres de Lie,” Hermann, Paris, (1968).
Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl. 6 (1957), 111–244.
Gatti, V., Viniberghi, E., Spinors of a 13-domensional space. Adv. Math. 30 (1978), 137–155.
Kac, V., Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190–213.
Kac, V., “Infinite dimensional Lie algebras, ” Birkhäuser Verlag, Basel-Boston, (1983).
Kostant, B., The principle three-dimensional subgroup and the Betti numbers of a complex simple Lie algebra, Amer. J. Math. 81 (1959). 973–1032.
Springer, T. A., Steinberg, R., “Conjugacy classes: Seminar on algebraic groups and related finite groups, ” Springer Lecture Notes. Springer Verlag, Berlin-Heidelberg-New York, 131 (1970).
Littelmann, P., Koreguläre und äquidimensionale Darstellungen. J. Algebra 123 (1989), 193–222.
Vinberg, E. B., On the classification of the nilpotent elements of a graded Lie algebra, Math. USSR-Izv. 10 (1976), 463–495.
Vinberg, E. B., The Weyl group of a graded Lie algebra. Sov. Math. Dokl. 16 (1975), 1517–1520.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Littelmann, P. (1996). An effective method to classify nilpotent orbits. In: González-Vega, L., Recio, T. (eds) Algorithms in Algebraic Geometry and Applications. Progress in Mathematics, vol 143. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9104-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9104-2_12
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9908-6
Online ISBN: 978-3-0348-9104-2
eBook Packages: Springer Book Archive