Characterizations of extra-invariant spaces under the left translations on a Lie group

In the context of a connected, simply connected nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the range functions. Consequently, the theory is valid for the Heisenberg group Hd,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^d,$$\end{document} a 2-step nilpotent Lie group.