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The Poisson transform and representations of a free group. (English) Zbl 0532.43006
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 $$f\in L^ 2(\Omega)\subset {\mathcal F}'(\Omega)$$, the Poisson transform $$P^ zf$$ is defined by $$(P^ zf)(x)=\int p^ z(x,\omega)f(\omega)d\mu(\omega).$$ Convolution of $$L^ 2(\Omega)$$ with the simple random walk $$\mu_ 1=p_ 1(\mu)$$ on G is an analogue of the Laplace-Beltrami-operator and one is concerned with its eigenvectors. In fact, the principal results of the paper now are as follows.
Theorem A: If $$z\not\in i\pi(\log R)^{-1}.{\mathbb{Z}},$$ then for an $$L^ 2(G)$$-function $$\Phi$$ one has $$\Phi *\mu_ 1=(R+1)^{-1}(R^ z+R^{1- z})\Phi$$ iff $$\Phi$$ is the Poisson transform of a martingale on $$\Omega$$.
Theorem B: For $$s\in]0,1[+i{\mathbb{R}}$$ the operators $$\pi_ z(x)$$ are uniformly bounded on $$H_ z(\Omega)$$ independently of x in G. -
Since G may be identified with a subgroup of $$Gl(2,{\mathbb{Q}}_ p)$$, applications on the representation theory of the latter group are given and some results of P. Sally are reproved.
Reviewer: K.H.Hofmann

##### MSC:
 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 22D40 Ergodic theory on groups 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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##### References:
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