This note provides new criteria on a unimodular group G and a discrete series representation (π, H_π) of formal degree d_π> 0 under which any lattice Γ ≤ G with vol(G/Γ)dπ≤1 (resp. vol(G/Γ)dπ≥1) admits g∈ H_π such that π(Γ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian–Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C^⁎-module. An important ingredient in the approach is that twisted group C^⁎-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C^⁎-algebra has strict comparison of projections.