The contraction of K to \=NM
type: Item
instance of: scholarly article
title: The contraction of K to \=NM
zbMATH Open document ID: 0588.22011
DOI: 10.1016/0022-1236(85)90092-8
author: Anthony H. Dooley
author: Fulvio Ricci
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: Let \(G=KAN\) be a connected semisimple Lie group of real rank one. Let \(H\in {\mathfrak a}^+\) be the unique element of norm 1. Define the dilation \(A_ t: G\to G\) \((t>0)\) and the contraction \(\pi_ t: \bar NM\to K\) \((t>0)\) as the following: \[ A_ t(g)=\exp (tH)\cdot g\cdot \exp (- tH),\quad g\in G,\quad \pi_ t(\bar nm)=k(A_ t(\bar n))\cdot m,\quad \bar n\in \bar N,\quad m\in M. \] The maps \(\pi_ t\) are diffeomorphisms onto a dense submanifold of K. Let \(X=G/K\) and \(\Omega =K/M\). The group K is the stabilizer of K-orbits in the symmetric space X and the group \(\bar NM\) is the stabilizer of every horocycle of the variety \(\Omega\). The group \(\bar NM\) is considered as a limit of the compact group K under the continuous family \((\pi_ t\); \(t>0)\) of contractions. Holomorphically induced irreducible unitary representations of the group \(\bar NM\), \(\sigma\) 's, and those of the group K, \(\rho\) 's, are constructed. The former are shown to be increasing limits of images under the maps \(\pi_ t\) of the latter, and using these facts, matrix coefficients of the representation \(\sigma\) are approximated by those of \(\rho\).
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 3944172
zbMATH Keywords: connected semisimple Lie group
zbMATH Keywords: real rank one
zbMATH Keywords: dilation
zbMATH Keywords: contraction
zbMATH Keywords: diffeomorphisms
zbMATH Keywords: dense submanifold
zbMATH Keywords: symmetric space
zbMATH Keywords: horocycle
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: matrix coefficients
reviewed by: Yasuhiro Asoo
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90092-8
OpenAlex ID: W2052874959
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description: scientific article; zbMATH DE number 3944172