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.