Let \(G\) be a second-countable unimodular locally compact group and let \((\pi, \mathcal{H}_\pi)\) be an irreducible projective unitary representation of \(G\). The paper studies sufficient conditions for the existence of so-called frames and Riesz sequences, where for a lattice \(\Gamma \leq G\), the \(\Gamma\)-orbit \(\pi(\Gamma)g\) of \(g \in \mathcal{H}_\pi\) is said to be a frame if \[ A \|f\|_{\mathcal{H}_\pi}^2 \leq \sum_{\gamma \in \Gamma} |\langle f, \pi(\gamma)g\rangle|^2 \leq B \|f\|_{\mathcal{H}_\pi}^2, \qquad \forall f \in \mathcal{H}_\pi \] for suitable constants \(A,B > 0\). The orbit \(\pi(\Gamma)g\) is called a Riesz sequence if for some \(A,B > 0\): \[ A \|c\|_{\ell^2}^2 \leq \| \sum_{\gamma} c_\gamma \pi(\gamma)g\|_{\mathcal{H}_\pi}^2 \leq B \|c\|_{\ell^2}^2, \qquad \forall c \in \ell^2(\Gamma). \] Suppose that \(\pi\) is a discrete series representation of formal dimension \(d_\pi > 0\) with trivial projective kernel. Assume further that any conjugacy class in \(G\) is pre-compact in \(G\) and that either \(G\) is locally connected or \(\Gamma\) is co-compact in \(G\). The main result of the paper is that under these assumptions, there exists a \(\Gamma\)-orbit \(\pi(\Gamma)g\) in \(\mathcal{H}_\pi\) which is a frame if \(\mathrm{vol}(G/\Gamma)d_\pi \leq 1\), and there exists an orbit \(\pi(\Gamma)g\) which is a Riesz sequence if \(\mathrm{vol}(G/\Gamma)d_\pi \geq 1\).
Assuming that \(G\) is \(1\)-connected, the condition that any conjugacy class in \(G\) is pre-compact is in particular satisfied if \(G\) is an exponential solvable Lie group, a complex-analytic Lie group or a semisimple Lie group without compact factors.