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

Borel, Armand

Ann. Math. (2) 72, 179-188 (1960).

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:

22E46

Semisimple Lie groups and their representations

Keywords:

connected, real, semisimple Lie group without compact components; stable subalgebra; centralizer