Near-Parseval orbit frames for irreducible unitary representations: from mixing and expansion

Abstract

We establish an abstract storage theorem for projectively $C_0$ unitary representations admitting a pair of one-parameter subgroups with conjugation escape. It produces, for every $0<\epsilon <1$, a single orbit whose sampling set is relatively separated modulo the projective kernel and whose frame bounds are $(1-\epsilon {)}^2$ and $(1+\epsilon {)}^2$. Two applications are obtained from the same mechanism. First, every nonabelian exponential Lie group admits a near-Parseval frame of left translates for its left regular representation. Consequently, a positive-dimensional exponential Lie group is an FT group if and only if it is nonabelian; in particular, the three-dimensional Heisenberg group admits such a frame. Second, every infinite-dimensional irreducible unitary representation of an exponential Lie group admits a near-Parseval discrete orbit frame. When the effective projective quotient is abelian, the orbit may be chosen to be an orthonormal basis arising from a transported Weyl lattice.

36 pages, 4 figures. Appendix A documents the exact scope of a partial Lean 4/Mathlib certification; a Lean source archive is included as an ancillary file