Geometric harmonic analysis on homogeneous Siegel domains

主要研究结果如下：1.引入了一族由可容许线性形式参数化为参数的Cayley变换，并证明了每个成员将Siegel域以双调和双全纯的方式映射到有界域上。这个系列包括Penney在1996年引入的Cayley变换以及首席调查员在2001年引入的Cayley变换。这一结果已经作为研究论文发表。2.我们证明了Poisson-hua核的调和性与区域的对称性是等价的。此外，即使我们以标准的方式改变Siegel域的Koehler度量，Poisson-hua核是调和的一个充要条件是，除了域的对称性外，所考虑的度量是Bergman度量的正数倍。这一结果也已经作为研究论文发表了。3.我们已经证明了解析连续伪逆映射保持…更多的管域当且仅当定义管域的凸锥是对称的。在实际定理中，伪逆映射带有一个参数，从而使该定理更具一般性。这一结果被作为研究论文发表(与博士生Chifune Kai联合发表的论文)。4.我们研究了在上述1中提到的Cayley变换下，均匀管域的像的凸性。Cayley变换和对偶Cayley变换的象凸的充要条件是管域对称。通过考虑Cayley变换族，以更一般的方式证明了实际定理。5.我们发现了一系列非对称开凸锥，它们的基本相对不变量的次数为1，2，…，r，其中r是开凸锥的秩数。此外，在与Ishi共同研究的过程中，我们证明了每个基本相对不变量都表现为复杂化族的右乘法算子的确定式的一个不可约因子。到目前为止，基本的相对不变性仅仅是作为算法的结果而获得的。我们目前正在就这一结果撰写一篇联合论文。较少

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