When is a reductive group*-invariant?

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.