Advertisement

Periodica Mathematica Hungarica

, Volume 26, Issue 1, pp 65–75 | Cite as

When is a reductive group*-invariant?

  • Z. Magyar
Article
  • 16 Downloads

Abstract

LetG be a real reductive Lie group, i.e., a Lie group whose Lie algebra is the direct product of a commutative and a semi-simple algebra. LetG0 be the unit component ofG. We analyze the following question: if Φ is a continuous linear representation ofG over a finite dimensional complex vector spaceV then when can we find a scalar product onV so that the group Φ(G) become*-invariant with respect to it? In particular, ifG/G0 is finite then we show that this is the case if and only if the same holds for the connected subgroup corresponding to the center ζ of the Lie algebra ofG and the latter condition is very easy to describe in terms ofdΦ/ζ. We discuss some related questions such as the relation between Cartan decompositions ofG and polar decompositions of Φ(G), the description of the closure of Φ(G), etc.

Mathematics subject classification numbers, 1991

Primary 22F46 

Key words and phrases

Reductive Lie group finite dimensional representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ada]J. F. Adams:Lectures on Lie Groups, Benjamin, New York, Amsterdam, 1969.Google Scholar
  2. [Bou]N. Bourbaki:Éléments de Mathématique, Groupes et Algèbres de Lie, Hermann, Paris, Chap. I: 1960, Chap. II–III: 1972, Chap. IV–VI: 1968, Chap. VII–VIII: 1975; Chap. IX: Masson, Paris, 1982.Google Scholar
  3. [Dy-O]E. B. Dynkin andA. L. Onishchik: Compactnye gruppy Li v celom (Compact global Lie groups),Usp. Mat. N. 10 (1955), 3–74.Google Scholar
  4. [H-Sch]M. Hausner andJ. T. Schwartz:Lie Groups; Lie Algebras, Gordon and Breach, New York, London, Paris, 1968.Google Scholar
  5. [Hel]S. Helgason:Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, London, 1978.Google Scholar
  6. [Kna]A. W. Knapp:Representation Theory of Semisimple Groups, Princeton University Press, Princeton, 1986.Google Scholar
  7. [Mag1]Z. Magyar: On the classification of real semi-simple Lie algebras,Acta Math. Hung. 54 (1989), 99–134.Google Scholar
  8. [Mag2]Z. Magyar:Continuous Linear Representations, North-Holland Math. Studies 168, Amsterdam, 1992.Google Scholar

Copyright information

© Akadémiai Kiadó 1993

Authors and Affiliations

  • Z. Magyar
    • 1
  1. 1.Mathematical InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations