The Poisson transform and representations of a free group
type: Item
instance of: scholarly article
title: The Poisson transform and representations of a free group
zbMATH Open document ID: 0532.43006
DOI: 10.1016/0022-1236(83)90019-8
author: Anna Maria Mantero
author: Anna Zappa
published in: Journal of Functional Analysis
publication date: 1983-01-01T00:00:00Z
review text: Let \(G=F(X)\) be the free group on the finite set X with \(r>1\) elements and \(F_ n(X)\) the subset of reduced words of length n; then \(F_ n(X)\) has \(rR^{n-1}\) elements with \(R=2r-1\). For \(m\leq n\) let \(p_{nm}:F_ n(X)\to F_ m(X)\) the projection onto the first m letters; then the inverse limit \(\Omega =\lim F_ n(X)\) of this inverse system is a Cantor set (which may be identified with the space of reduced words of infinite length). Let \(p_ n:\Omega \to F_ n(X)\) be the limit maps and \({\mathcal F}_ n(\Omega)\) the vector space of functions of the form \(f{\mathbb{O}}p_ n\) and let \({\mathcal F}(\Omega)\) be the direct limit \(\cup {\mathcal F}_ n(\Omega)\). If one considers \({\mathcal F}_ n(\Omega)\) as a (finite dimensional) Hilbert space \((L^ 2\) relative to normalized counting measure on \(F_ n(X))\), then \({\mathcal F}'_ n(\Omega)\) may be identified with \({\mathcal F}_ n(\Omega)\). The dual \({\mathcal F}'(\Omega)\) is the projective limit of the \({\mathcal F}_ n(\Omega)\) and its elements \((f_ n)\) are the martingales on \(\Omega\). On \(\Omega\) one considers the unique probability measure \(\mu\), for which \(p_ n(\mu)\) is normalized counting measure on \(F_ n(X)\). The group G acts on \(\Omega\) on the left in such a fashion that \(\mu\) is quasiinvariant. The Poisson kernel \(p(x,\omega)=d\mu(x^{-1}\omega)/d\mu(\omega)\) is a power of the integer R whose exponent is an integer depending on x and \(\omega\). For each complex number z one now defines a representation of G in \({\mathcal F}(\Omega)\) via \((\pi_ z(x)f)(\omega)=p^ z(x,\omega)f(x^{- 1}\omega)\). This representation extends to one in \({\mathcal F}'(\Omega)\). For \(z\in]0,1[+i\pi(\log R)^{-1}.{\mathbb{Z}}\) the representation \(\pi_ z\) (complementary series!) is unitary on a suitable Hilbert space \(H_ z(\Omega)\) with an inner product obtained from \(L^ 2(\Omega)\) via a certain explicitly constructed intertwining kernel operator. For \(f\in L^ 2(\Omega)\subset {\mathcal…
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 43A30
zbMATH DE Number: 3844353
zbMATH Keywords: free group
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Laplace Beltrami operator
zbMATH Keywords: Poisson kernel
zbMATH Keywords: uniform boundedness
zbMATH Keywords: representation
zbMATH Keywords: martingale
zbMATH Keywords: intertwining operator
zbMATH Keywords: complementary series
MaRDI profile type: Publication
full work available at URL: 0022-1236(83)90019-8
OpenAlex ID: W2063937230
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cites work: Q4055870
cites work: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)
description: scientific article; zbMATH DE number 3844353