The Cayley transform and uniformly bounded representations

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KA N ̄ and Bruhat big cell NMA N ̄ . Then the space G/MA N ̄ may be (almost) identified with N and with K/ M, and these identifications induce the (generalised) Cayley transform C : N→K/M . We show that C is a conformal map of...

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Bibliographic Details
Published in:Journal of functional analysis 2004-08, Vol.213 (2), p.241-269
Main Authors: Astengo, F, Cowling, M, Di Blasio, B
Format: Article
Language:English
Subjects:
Cayley transform
Simple Lie groups of real rank one
Uniformly bounded representations
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Summary:Let G be a simple Lie group of real rank one, with Iwasawa decomposition KA N ̄ and Bruhat big cell NMA N ̄ . Then the space G/MA N ̄ may be (almost) identified with N and with K/ M, and these identifications induce the (generalised) Cayley transform C : N→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 H α(N) and H α(K/M) . We use this to construct uniformly bounded and slowly growing representations of G.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2003.12.009