For a second countable locally compact group G and a closed abelian subgroup H, we give a range function classification of closed subspaces in L 2 (G) invariant under left translation by H. For a family A ⊆ L 2 (G), this classification ties with a set of conditions under which the translations of A 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 ⊆ G, we produce a measure on the space Γ\G of right cosets that gives a measure space isomorphism G ∼ = Γ ×Γ\G. Outside of the group setting, we consider a more general problem: for a measure space X and a Hilbert space H, we investigate conditions under which a family of functions in L 2 (X; H) multiplies with a basis-like system in L 2 (X) to produce a continuous frame or a Riesz sequence in L 2 (X; H). Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernández et al. in [25].

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of abelian distance-regular antipodal covers of the complete graph.

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

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