Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree |
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Authors: | G Kuhn A Vershik |
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Institution: | (1) Dipartimento di Matematica Università degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20123 Milan, Italy;(2) St. Petersburg branch Mathematical Institute of Russian Academy of Science, St. Petersburg, 191011, Russia |
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Abstract: | Let G=A
ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|=0=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH
in some representation space H
. In contrast with the case of PGL(2,), the representation is not irreducible.Let
o
(g) be the derivative of the spherical function corresponding to the complementary series of A
ut(T). We show that –d(e,ge) and
o
(g) come from cohomologous cocycles. Moreover,
o
is associated to one of the two (irreducible) special representations of A
ut(T). |
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Keywords: | conditionally positive definite function 1-cocycle complementary series representations |
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