On the rational closure of connected closed subgroups of connected simply connected nilpotent Lie groups (Q2274799)

Let \(G\) be a connected simply connected nilpotent Lie group and \(\Gamma\) be a lattice (discrete uniform subgroup) in \(G\). A connected Lie subgroup \(H \subset G\) (it is always closed in such \(G\)) is called \(\Gamma\)-rational if \(H \cap \Gamma\) is a lattice in \(H\). For any connected Lie subgroup \(H \subset G\) the smallest \(\Gamma\)-rational normal connected Lie subgroup \(\mathcal{N}_G^\Gamma(H)\), containing \(H\), is called the normal \(\Gamma\)-rational closure of \(H\). The smallest \(\Gamma\)-rational connected Lie subgroup \(\mathcal{A}_G^\Gamma(H)\), containing \(H\), is called the \(\Gamma\)-rational closure of \(H\). In this article subgroups \(\mathcal{N}_G^\Gamma(H)\) and \(\mathcal{A}_G^\Gamma(H)\) are described explicitly. This is done using the subgroup \(\mathcal{I}(H, \Gamma)\) which is the identity component of the closure of the subgroup generated by \(H\) and \(\Gamma\). It is proved that \(\mathcal{N}_G^\Gamma(H) = \mathcal{I} (H, \Gamma)\). The main result of this paper is the following. Let \(G\) be a connected simply connected nilpotent Lie group and \(\Gamma\) be a lattice in \(G\). Then for any connected Lie subgroup \(H\) we have \(\mathcal{A}_G^\Gamma(H) = R_\infty(H)\), where \(R_\infty(H)\) is defined using some descending series of Lie \(\Gamma\)-rational subgroups. Two examples for 3-dimensional and 4-dimensional Lie groups \(G\) are presented. Then some applications for nilflows are proved. In particular, a criterion for the ergodicity of the action of \(H\) on \(G/\Gamma \) is proved. Also a characterization of the irreducible unitary representations of \(G\), for which the restrictions to \(\Gamma \) remain irreducible, is given.