by Kuhn, G.; Vershik, A.

Let G=A
ut(T) be the group of automorphisms of a homogeneous tree and let d(v,g⋅v) denote the natural tree distance. Fix a base vertex e in T. The function φ^μ(g)=exp(−μd(e,g⋅e)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator dφ^μ/dμ|_μ=0=log(φ) is conditionally positive definite but not positive definite. Hence, log(φ) corresponds to a nontrivial cocycle β(g): G→H
_π in some representation space H
_π. In contrast with the case of PGL(2,ℝ), the representation π is not irreducible.Let ψ
_o
(g) be the derivative of the spherical function corresponding to the complementary series of A
ut(T). We show that −d(e,g⋅e) and ψ
_o
(g) come from cohomologous cocycles. Moreover, ψ
_o
is associated to one of the two (irreducible) special representations of A
ut(T).

DOI: 10.1023/A:1025163802191
Print publication date: 8/1/2003
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