## Density properties for certain subgroups of semi-simple groups without compact components.(English)Zbl 0094.24901

Suppose $$G$$ is a connected, real, semisimple Lie group without compact components. A subgroup $$H$$ of $$G$$ may be such that (i) $$H$$ is discrete and the measure of $$G/H$$ is finite, or (ii) for every neighbourhood $$U$$ of the identity in $$G$$ and every element $$g\in G$$ there is an integer $$n$$ so that $$g^n\in UHU$$. (i) implies (ii), and is, of course, the more interesting property. For the author’s purposes, however, it is convenient to assume (ii) alone. The author then shows that if a subalgebra of the Lie algebra of $$G$$ is stable under $$H$$ it is an ideal. Among other corollaries he deduces that (i) the centralizer of $$H$$ is the centre of $$G$$ and (ii) if $$\rho$$ is a representation of $$G$$ in a real (complex) vector space then every element of $$\rho(G)$$ is a real (complex) linear combination of elements of $$\rho(H)$$.
Reviewer: R. P. Langlands

### MSC:

 2.2e+47 Semisimple Lie groups and their representations
Full Text: