Representations of double coset hypergroups and induced representations. (English) Zbl 0844.43007
The author introduced the concept of induced representations of hypergroups on the basis of the abstract induction for \({\mathcal C}^*\)-algebras due to Rieffel in a series of papers. This inducing process is more complicated than in the group case. For instance, even for commutative hypergroups, there may exist representations which are not inducible. On the other hand, induction has some nontrivial applications to the duality theory of commutative hypergroups; see the author [Math. Z. 211, No. 4, 687-699 (1992; Zbl 0776.43002) and Monatsh. Math. 116, No. 3-4, 245-262 (1993; Zbl 0793.43004)] and the author and the reviewer [Forum Math. 7, No. 5, 543-558 (1995; Zbl 0832.43006)]. The paper under review investigates the connections between the induction process and forming double coset hypergroups. This generalizes connections between induction and proper hypergroup homomorphisms and is important in particular in the case of Gelfand pairs \((G,H)\). A major part of the paper is devoted to a class of examples, namely the Gelfand pair \((SL (2,F)\), \(SL (2,R))\) with \(F\) a local field and \(R\) the ring of integers in \(F\). As well known from papers of P. Cartier and others, the associated double coset hypergroup is isomorphic to the discrete double coset hypergroup coming from the automorphism group of a homogeneous tree and may thus be regarded as a polynomial hypergroup on \(\mathbb{N}_0\).
Reviewer: M.Voit (Tübingen)
MSC:
| 43A62 | Harmonic analysis on hypergroups |
| 22D30 | Induced representations for locally compact groups |
| 22D15 | Group algebras of locally compact groups |
| 22E35 | Analysis on \(p\)-adic Lie groups |
Keywords:
hypergroups; representations; duality theory; homomorphisms; Gelfand pairs; Gelfand pair; double coset hypergroup; homogeneous tree; polynomial hypergroupReferences:
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