Linear dependency of translations and square-integrable representations

Abstract

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2\left(G\right)$ have linearly independent left translations. In particular, we establish some results for the case when $G$ has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.