Induced representations and hypergroup homomorphisms
Induced representations on hypergroups were introduced by the author [in Math. Z. 211, No. 4, 687-699 (1992; Zbl 0776.43002)] where some difficulties appear which are unknown for groups. For instance, there may exist non-inducible characters, and even the trivial one may fail to be inducible for commutative hypergroups. In the present paper, the author introduces Weil subhypergroups and Weil homomorphisms and studies relations to induced representations. According to the author, a subhypergroup \(H\) of a hypergroup \(K\) is called Weil if \(T_ H(f)(x * H) := \int_ H f(x*t)d\omega_ H(t)\) leads to a well-defined mapping \(T_ H: C_ \infty(K) \to C_ \infty(K/H)\). Moreover, a hypergroup homomorphism is called Weil if its kernel is a Weil subhypergroup. All subgroups and all compact or supernormal subhypergroups have the Weil property, but there exist examples which fail to be Weil. A Weil decomposition formula for Weil homomorphisms is given in this paper. This in particular proves that lifting of representations via Weil homomorphisms is compatible with the inducing process. Moreover, it follows for supernormal subhypergroups \(H\) of a commutative hypergroup \(K\) that characters on \(H\) are inducible to \(K\) if and only if they can be extended to characters on \(K\).
scientific article; zbMATH DE number 508785
scientific article