The Cayley transform and uniformly bounded representations

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KAN and Bruhat big cell NMAN. Then the space G/MAN may be (almost) identified with N and with K/M; and these identifications induce the (generalised) Cayley transform C from N into K/M. We show that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplication by appropriate powers of the Jacobian, induces isomorphisms of Sobolev spaces on N and K/M. We use this to construct bounded and slowly growing representations of G.