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 ${\mathbb{H}}^d$, a 2-step nilpotent Lie group.