The Poisson transform and representations of a free group
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.
scientific article; zbMATH DE number 3844353