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UBC Theses and Dissertations

Semisimple symmetric spaces Rossmann, Wulf


The thesis contains a structure theory for semisimple symmetric spaces with applications to related problems in analysis. It is shown how a root system can be associated with a semisimple symmetric space (abr. "s.s, space"). Analogues of the Cartan, Iwasawa, Bruhat decompositions of a semisimple Lie group are established for arbitrary s.s. spaces. The "conical dual" of a s.s. space is introduced. The representation of a semisimple transformation group on L²-functions on the conical dual is decomposed as a direct integral of principal series representations. The algebra of invariant differential operators on the conical dual is found to be commutative and its eigenfunctions are determined. Intertwining operators between compactly supported functions on a s.s. space and functions on its conical dual, and v.v., are defined. The definition is extended to a larger class of functions by an investigation of the convergence properties of these operators. An application to the harmonic analysis on SO(p,q)₀/SO(p,q-1)₀ is indicated.

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