Multiplicity one for representations corresponding to spherical distribution vectors of class

Helminck, G.F. and Helminck, A.G. (2002) Multiplicity one for representations corresponding to spherical distribution vectors of class . [Report]

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 Abstract: In this paper one considers a unimodular second countable locally compact group and the homogeneous space , where is a closed unimodular subgroup of . Over complex vector bundles are considered such that acts on the fibers by a unitary representation with closed image. The natural action of on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class . The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for a symmetric -variety , with a locally compact field of characteristic not equal to two. Here is a reductive algebraic group defined over and is the fixed point group of an involution of defined over . It is shown then that the natural representation of on the Hilbert space is multiplicity free if is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair is a generalized Gelfand pair. Here and are suitable algebraic groups defined over . Item Type: Report Faculty: Electrical Engineering, Mathematics and Computer Science (EEMCS) Research Group: Link to this item: http://purl.utwente.nl/publications/65852 Export this item as: BibTeXEndNoteHTML CitationReference Manager

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