Random walks on free products

Let G = double-ended-daggar (j = 1)q + 1G(n(j)-l) be product of q + 1 finite groups each having order n(1) + 1 and let mu 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 mu in C(reg)* (G) 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 mu in C(reg)* (G) in several cases. A family of irreducible representations of G, parametrized on the continuous spectrum of mu. is here presented. Finally, we shall get a decomposition of the regular representation of G by means of the Green function of mu and the decomposition is into irreducibles if and only if there are no true eigenspaces for mu