On sufficient density conditions for lattice orbits of relative discrete series

Abstract

This 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.

A Groupoid Approach to Interacting Fermions

“…The mathematical aspects of patterns in non-abelian groups have been steadily attracted an interest from the mathematics community in the past few years [11,12,13,14,15,16,4]. In fact, during the completion of our work, Enstad, Raum and van Velthoven posted the works [27,28] on frame theory for patterns in locally compact second countable (lcsc) groups. These works not only cleared for us all the technical challenges, but they also supplied the natural mathematical framework we were looking for.…”