, Volume 88, Issue 1, pp 1-24

Representations of double coset hypergroups and induced representations

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Abstract

The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair \(SL(2,\mathfrak{K})//SL(2,R)\) (where \(\mathfrak{K}\) are a local field andR its ring of integers) as a polynomial hypergroup on ℕ0 and characterize the (proper) subset of its dual consisting of extensible representations.