Abstract
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 φμ(g)=exp(−μd(e,g⋅e)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator dφμ/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).
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Kuhn, G., Vershik, A. Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree. Algebras and Representation Theory 6, 333–352 (2003). https://doi.org/10.1023/A:1025163802191
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DOI: https://doi.org/10.1023/A:1025163802191