  Let $N$ be the Heisenberg group. We consider left-invariant multiplicity free
subspaces of $L^2(N)$. We prove a necessary and sufficient density condition in
order that such subspaces possess the interpolation property with respect to a
class of discrete subsets of $N$ that includes the integer lattice. We exhibit
a concrete example of a subspace that has interpolation for the integer
lattice, and we also prove a necessary and sufficient condition for shift
invariant subspaces to possess a singly-generated orthonormal basis of
translates.
