REPRESENTATIONS OF DOUBLE COSET HYPERGROUPS AND INDUCED REPRESENTATIONS

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, K)//SL(2, R) (where R 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.