Complex integral geometry

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AbstractGindikinThe focus of the proposal is the development of a method of complexhorospheras for real affine symmetric spaces, including real semisimpleLie groups. We anticipate to apply this method to the construction of themodels of series of representations and to the decomposition of the regularrepresentations on series on the language of Hardy spaces ofcohomology. As one of application we hope to give an integral geometrical proof of the product-formula for c-function of Harishi-Chandra. Another direction of my research is the investigation of geometrical and analytic properties ofthe crowns of Riemann symmetric spaces - their canonical Steinneighborhoods. Integral geometry is a branch of geometrical analysis whichinvestigates integral transforms associated with different geometricalstructures. The principal challenge is to find a geometrical setting, moregeneral than the group one, where it is possible to develop a non-abelianharmonic analysis. Another direction considers complex constructionsresponsible for the geometry of real affine symmetric spaces. Suchphenomena have roots in classical geometry of Poncelet and Pluecker.