  We study Koopman and quasi-regular representations corresponding to the
action of arbitrary weakly branch group G on the boundary of a rooted tree T.
One of the main results is that in the case of a quasi-invariant Bernoulli
measure on the boundary of T the corresponding Koopman representation of G is
irreducible (under some general conditions). We also show that quasi-regular
representations of G corresponding to different orbits and Koopman
representations corresponding to different Bernoulli measures on the boundary
of T are pairwise disjoint. This gives two continual collections of pairwise
disjoint irreducible representations of a weakly branch group. Another
corollary of our results is triviality of the centralizer of G in various
groups of transformations on the boundary of T.
