# Representations of double coset hypergroups and induced representations

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DOI: 10.1007/BF02567801

- Cite this article as:
- Hermann, P. Manuscripta Math (1995) 88: 1. doi:10.1007/BF02567801

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## Abstract

The principal goal of this paper is to investigate the representation theory of double coset hypergroups. If*K=G//H* is a double coset hypergroup, representations of*K* can canonically be obtained from those of*G*. However, not every representation of*K* originates from this construction in general, i.e., extends to a representation of*G*. Properties of this construction are discussed, and as the main result it turns out that extending representations of*K* is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation of*K* always admits an extension to*G*. Furthermore, we realize the Gelfand pair\(SL(2,\mathfrak{K})//SL(2,R)\) (where\(\mathfrak{K}\) are a local field and*R* its ring of integers) as a polynomial hypergroup on ℕ_{0} and characterize the (proper) subset of its dual consisting of extensible representations.