Geometric harmonic analysis on homogeneous Siegel domains

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The main research results are listed as follows.1.We have introduced a family of Cayley transforms parametrized by admissible linear forms, and proved that every member maps the Siegel domain onto a bounded domain birationally and biholomorphically. This family contains the Cayley transform introduced by Penney in 1996 as well as the Cayley transform introduced by the head investigator in 2001. This result is already published as a research paper.2.We have proved that the harmonicity of the Poisson-Hua kernel is equivalent with the symmetry of the domain. Moreover, even if we vary the Koehler metric of the Siegel domain in the standard way, a necessary and a sufficient condition for the Poisson-Hua kernel to be harmonic is that, in addition to the symmetry of the domain, the metric considered is a positive number multiple of the Bergman metric. This result, too, is already published as a research paper.3.We have established that the analytically continued pseudoinverse map preserves th … More e tube domain if and only if the convex cone defining the tube domain is symmetric. In the actual theorem, the pseudoinverse map is equipped with a parameter, so that the theorem is a little more general. The result is published as a research paper (joint paper with a doctor course student Chifune Kai).4.We investigated the convexity of the images of homogeneous tube domains under the Cayley transforms mentioned in 1 above. A necessary and sufficient condition for the images of the Cayley and the dual Cayley transforms to be convex is that the tube domains are symmetric. The actual theorem is proved in a little more general way by considering the family of Cayley transforms. The result is published as a research paper (joint with a doctor course student Chifune Kai).5.We have found a series of non-symmetric open convex cones such that the degrees of the associated basic relative invariants are 1,2,...,r, where r is the rank of the open convex cone. Moreover in the course of joint research with Ishi we have shown that every basic relative invariant appears as an irreducible factor of the detgerminant of the right multiplication operator of the complexified clan. So far basic relative invaraats were obtained as a mere algorithmic resultant. We are currently writing a joint paper about this results. Less