Density conditions with stabilisers for lattice orbits of discrete series representations

TL;DR

This paper offers simplified proofs for density conditions of frames and Riesz sequences in lattice orbits of discrete series representations, extending known results for Bergman kernels and Lie groups.

Contribution

It introduces elementary proofs that extend and simplify existing density conditions involving stabilisers in the context of discrete series representations.

Findings

Elementary proofs for density conditions

Extension of known results to broader settings

Simplification of previous proofs

Abstract

This note provides elementary proofs for necessary density conditions for frames and Riesz sequences in the lattice orbit of a discrete series representation that involve the projective stabiliser of the vector. The presented approach extends and simplifies known density conditions for Bergman kernels and lattices in semisimple Lie groups.