Continuous Families of Quasi-Regular Representations of Solvable Lie Groups

We generalize a result about two-step nilpotent Lie groups by Carolyn Gordon and He Ouyang: Let {Γt}t ≥ 0 be a continuous family of uniform discrete subgroups of a simply connected Lie group G such that the quasi-regular representations of G on L2(Γt\G) are unitarily equivalent. Then if G is solvable with only real roots, there exists a unique continuous family {φt}t ≥ 0 of Γ0-almost inner automorphisms of G such that Γt = Φt(Γ0). An even stronger result is true for exponentially solvable Lie groups (not necessarily with real roots) whose nilradical is abelian and of codimension one, at least if we assume that {Γt}t ≥ 0 is not only continuous but C1. Here the Γt are related not only by almost inner, but by inner automorphisms.