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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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