Subspaces of L2(G)L2(G) invariant under translation by an abelian subgroup

For a second countable locally compact group G and a closed abelian subgroup H  , we give a range function classification of closed subspaces in L2(G)L2(G) invariant under left translation by H  . For a family A⊆L2(G)A⊆L2(G), this classification ties with a set of conditions under which the translations of AA by H form a continuous frame or a Riesz sequence. When G   is abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on a new group-theoretic tool: for a closed subgroup Γ⊆GΓ⊆G, we produce a measure on the space Γ\GΓ\G of right cosets that gives a measure space isomorphism G≅Γ×Γ\GG≅Γ×Γ\G. Outside of the group setting, we consider a more general problem: for a measure space X   and a Hilbert space HH, we investigate conditions under which a family of functions in L2(X;H)L2(X;H) multiplies with a basis-like system in L2(X)L2(X) to produce a continuous frame or a Riesz sequence in L2(X;H)L2(X;H). Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernández et al. in [25].