On sufficient density conditions for lattice orbits of relative discrete series

Abstract

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