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Harmonic analysis for anisotropic random walks on homogeneous trees. (English) Zbl 0836.43019
Mem. Am. Math. Soc. 531, 68 p. (1994).
Let \(\Gamma\) be a homogeneous tree of order \(q + 1\). Assume that the edges are labelled with integers between 1 and \(q + 1\) such that the edges of each vertex are labelled differently. The automorphism group \(G\) of the labelled tree can be realized as the free product of \(q + 1\) copies of \(\mathbb{Z}_2\), and \(G\) can be identified with the set of all vertices of \(\Gamma\). Now choose \(p_1,\dots, p_{q+1} > 0\) with \(\sum p_i = 1\), and consider the associated nearest neighborhood random walk on \(G\) (or \(\Gamma\) respectively) which is symmetric and which depends on the labelling above. The associated convolution operator \(R\) on \(\ell^2(G)\) is continuous and commutes with the left regular representation \(\lambda\) of \(G\), i.e., for each generalized eigenspace \(H_\sigma\) of \(R\) one obtains a subrepresentation \(\pi_\sigma\) of \(\lambda\). The main purpose of these notes is to study these subrepresentations \(\pi_\sigma\) of \(\lambda\). For this, the spectrum of the operator \(R\) on \(l^2(G)\) is described completely, and \(l^2 (G)\) is written as a direct integral of generalized eigenspaces of \(R\). Moreover, based on the boundary of the tree, complementary series representations called \(\pi_\gamma\) and \(\pi^\pm_\gamma\) of \(G\) are determined. As a main result it turns out that (up to some exceptions) all representations \(\pi_\sigma\), \(\pi_\gamma\) and \(\pi^\pm_\gamma\) are irreducible and inequivalent. Finally, principal series representations \(\pi_\sigma\) for different random walks are always inequivalent.
These notes (which are based on the PhD-thesis of T. Steger) considerably generalize the case of isotropic random walks on \(\Gamma\) which was studied by A. Figà-Talamanca and M. A. Picardello [Harmonic Analysis on Free Groups. Lecture Notes in Pure and Applied Mathematics, Vol. 87, New York-Basel (1983; Zbl 0536.43001)]. In the isotropic unlabelled case (in which one has an abelian setting), the automorphism group \(G\) of \(\Gamma\) is the free group \(F_{q + 1}\); here the above results are closely related to the representation theory of the \(q\)-adic groups \(\text{PGL}(2, q)\). Moreover, as the authors point out, some material of these notes can also be found in K. Aomoto [J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 297-318 (1984; Zbl 0583.60068)].
Reviewer: M.Voit (Tübingen)

MSC:
43A90 Harmonic analysis and spherical functions
22D10 Unitary representations of locally compact groups
22E46 Semisimple Lie groups and their representations
20E05 Free nonabelian groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
22E40 Discrete subgroups of Lie groups
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