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Random walks on free products

M. Gabriella Kuhn — 1991

Annales de l'institut Fourier

Let G = * j = 1 q + 1 G n j + 1 be the product of q + 1 finite groups each having order n j + 1 and let μ be the probability measure which takes the value p j / n j on each element of G n j + 1 { e } . In this paper we shall describe the point spectrum of μ in C reg * ( G ) and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers n j . We also compute the continuous spectrum of μ in C reg * ( G ) in several cases. A family of irreducible representations of G , parametrized on the continuous spectrum of μ ,...

Restricting cuspidal representations of the group of automorphisms of a homogeneous tree

Donald I. CartwrightGabriella Kuhn — 2003

Bollettino dell'Unione Matematica Italiana

Let X be a homogeneous tree in which every vertex lies on q + 1 edges, where q 2 . Let A = A u t X be the group of automorphisms of X , and let H be the its subgroup P G L 2 , F , where F is a local field whose residual field has order q . We consider the restriction to H of a continuous irreducible unitary representation π of A . When π is spherical or special, it was well known that π remains irreducible, but we show that when π is cuspidal, the situation is much more complicated. We then study in detail what happens when the...

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