Let G be a simple Lie group of real rank one, with Iwasawa decomposition KA \bar N and Bruhat big cell NMA\bar N: Then the space G/MA \bar N may be (almost) identiﬁed with N and with K /M, and these identiﬁcations induce the (generalised) Cayley transform C : N \to 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 on K/M. We use this to construct uniformly bounded and slowly growing representations of G.
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