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)].

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 |