Decomposition and Admissibility for the Quasiregular Representation for Generalized Oscillator Groups

We analyze the co-normally induced quasiregular representation for two families of Lie groups: the 2d-oscillator groups N!SO(2d) where N is the free two-step nilpotent group on 2d generators, and the dilated 2d-oscillator groups N ! (SO(2d)× R+). We construct irreducible decompositions in both cases with explicit spectrum and intertwining operators, and in both cases we prove a Caldéron-type admissibility condition for multiplicity-free, quasiequivalent subrepresentations. We prove that in the case of the 2d-oscillator groups, the quasiregular representation has no admissible vectors, and for the dilated 2d-oscillator groups, we give an explicit construction for admissible vectors.