On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group $G$ and a discrete series representation $(\pi, \mathcal{H}_{\pi})$ of formal degree $d_{\pi} > 0$ under which any lattice $\Gamma \leq G$ with $\text{vol}(G/\Gamma) d_{\pi} \leq 1$ (resp. $\text{vol}(G/\Gamma) d_{\pi} \geq 1$) admits $g \in \mathcal{H}_{\pi}$ such that $\pi(\Gamma) g$ is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.