On sufficient density conditions for lattice orbits of relative discrete series

TL;DR

This paper establishes new density criteria for lattices in unimodular groups to admit frames or Riesz sequences via discrete series representations, with applications to exponential Lie groups.

Contribution

It introduces novel conditions on unimodular groups and discrete series representations ensuring the existence of frames or Riesz sequences for certain lattice densities.

Findings

Criteria for lattices with density ≤ 1 admit frames.

Criteria for lattices with density ≥ 1 admit Riesz sequences.

Applicable to all projective discrete series of exponential Lie groups.

Abstract

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.