The Cayley transform and uniformly bounded representations (Q1883591)

Let \(G\) be a simple Lie group of real rank one with Iwasawa decomposition \(KAN^{-}\) and Bruhat big cell \(NMAN^{-}\). The space \(G/MAN^{-}\) can (almost) be identified with \(N\) and \(K/M\), and the identifications induce the generalised Cayley transform. The authors show that the Cayley transform is a conformal map of Carnot-Caratheodory manifolds, and that composition with the Cayley transform and multiplication by appropriate powers of the Jacobian induces isomorphisms of the Sobolev spaces \(H^{\alpha}(N)\) and \(H^{\alpha}(K/M)\). The authors use this to construct uniformly bounded representations of \(G\) on the spaces \(H^{\alpha}(N)\) and \(H^{\alpha}(K/M)\) and slowly growing representations of \(G\).