Canonical semigroups of states and cocycles for the group of automorphisms of a homogeneous tree

Let G = script A signut(script T sign) be the group of automorphisms of a homogeneous tree and let d(v, g . u) denote the natural tree distance. Fix a base vertex e in script T sign. The function øμ (g) = exp(-μd(e, g . e)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator døμ |dμ|μ = log(ø) is conditionally positive definite but not positive definite. Hence, log(ø) corresponds to a nontrivial cocycle β (g): G → Hπ in some representation space H π. In contrast with the case of PGL(2, ℝ), the representation π is not irreducible. Let ψ0(g) be the derivative of the spherical function corresponding to the complementary series of script A signut(script T sign). We show that -d(e, g . e) and ψ0(g) come from cohomologous cocycles. Moreover, ψ0 is associated to one of the two (irreducible) special representations of script A signut(script T sign)