Maximal compact subgroups and the centre in an analytic group (Q2777717)

In the paper the centre, maximal tori and maximal compact subgroups of a simply connected analytic group are studied. Theorem. Let \(G\) be an analytic simply connected group, \(Z(G)\) its centre, \(R\) the radical of \(G\), and \(S\) a semisimple Levi factor of \(G.\) If \(R\) is nilpotent then \(Z(G)=(Z(G)\cap R)\times (Z(G)\cap S).\) An example of a simply connected group \(G\) for which \(Z(G)\cap R\) is not a direct factor of \(Z(G)\) is given. Theorem. Let \(G\) and \(G'\) be analytic groups, and let \(\alpha\) be a continuous homomorphism from \(G\) onto \(G'.\) Let \(H\) be an analytic subgroup of \(G\) containing \(\text{Ker }{\alpha}.\) Then \(H\) contains a maximal torus of \(G\) if and only if \({\alpha}(H)\) contains a maximal torus of \(G'.\) An analogous result is proved for maximal compact subgroups.