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Restrictions of the special representation of $$\text{Aut}(\text{tree}_ 3)$$ to two cocompact subgroups. (English) Zbl 0795.22004
This paper deals with a special problem in the harmonic analysis for groups acting on trees. Given a homogeneous tree $$T$$ of degree 3 (i.e., a connected infinite combinatorial graph with no loop and with three edges leaving each vertex), let $$G$$ be the automorphism group of $$T$$. With regard to its representation theory this group is quite analogous to the special linear group $$SL_ 2(\mathbb{R})$$; the irreducible unitary representations are classified. Given a discrete cocompact subgroup $$\Gamma$$ of $$G$$ the paper exhibits particular decompositions of the restriction to $$\Gamma$$ of the two special discrete series representations of $$G$$ for two specific choices of $$\Gamma$$.

##### MSC:
 22D10 Unitary representations of locally compact groups 22E40 Discrete subgroups of Lie groups 20E08 Groups acting on trees 22E35 Analysis on $$p$$-adic Lie groups 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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