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

Abstract

Let G be a connected simply connected nilpotent Lie group with discrete uniform subgroup $\Gamma$. A connected closed subgroup H of G is called $\Gamma$-rational if $H\cap \Gamma$ is a discrete uniform subgroup of H. For a closed connected subgroup H of G, let $\mathcal {I}(H, \Gamma )$ denote the identity component of the closure of the subgroup generated by H and $\Gamma$. In this paper, we prove that $\mathcal {I}(H, \Gamma )$ is the smallest normal $\Gamma$-rational connected closed subgroup containing H. As an immediate consequence, we obtain that $\mathcal {I}(H, \Gamma )$ depends only on the commensurability class of $\Gamma$. As applications, we give two results. In the first, we determine explicitly the smallest $\Gamma$-rational connected closed subgroup containing H. The second is a characterization of ergodicity of nilflow $(G/\Gamma , H)$ in terms of $\mathcal {I}(H, \Gamma )$. Furthermore, a characterization of the irreducible unitary representations of G for which the restriction to $\Gamma$ remain irreducible is given.