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

摘要

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