Complete reducibility and Zariski density in linear Lie groups (Q1818772)

Let \(G\) be a locally compact group, \(H\) a closed subgroup of \(G\) and \(\rho\) a continuous representation of \(G\) by bounded linear operators on a Banach space. The first problem this paper is concerned with is the relation between complete reducibility of \(\rho\) and its restriction to \(H\). It is shown that if \(\rho\) is finite dimensional and either \(G/H\) is compact or \(G/H\) has a finite invariant measure, then complete reducibility of \(\rho|H\) implies that \(\rho\) is completely reducible. It is mentioned that Furstenberg has constructed an example which shows that this conclusion does not remain true for infinite dimensional representations. However, the author proves that, even when \(\rho\) is infinite dimensional, complete reducibility of \(\rho \mid n\) implies that of \(\rho\) provided that \(G/H\) is compact and carries a finite invariant measure. The second topic dealt with is the Zariski density of \(H\) in \(G\). Exploiting an earlier density theorem of R. Mosak and the author, two different sets of conditions on \(G\) and \(H\) are elaborated each of which guarantees the Zariski density of \(H\). For instance, let \(G\) be a Zariski connected linear algebraic group defined over \(\mathbb{Q}\). Then \(G_\mathbb{Z}\), the group of integral points of \(G\), is Zariski dense in \(G\) precisely when \(G_\mathbb{R}\) is \(\mathbb{Q}\)-minimally almost periodic and the group of \(\mathbb{Q}\)-rational characters of \(G\) is trivial. Finally, combining the results on complete reducibility and on density, the author establishes the very satisfying result that when \(G\) is a linear algebraic group over \(\mathbb{Q},H\) an arithmetic subgroup of \(G\) and \(\rho\) a rational representation of \(G\), then \(\rho\) is completely reducible if and only if \(\rho\mid H\) is completely reducible.