Periodica Mathematica Hungarica

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

When is a reductive group*-invariant?

  • Z. Magyar


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 


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Copyright information

© Akadémiai Kiadó 1993

Authors and Affiliations

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

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