Geometric harmonic analysis on homogeneous Siegel domains

主要研究成果如下：1。引入了一类由可容许的线性形式参数化的Cayley变换，并证明了该类变换的每一个元都将Siegel域双全纯地映射到一个有界域上。这个家族包含了Penney在1996年提出的Cayley变换和首席调查员在2001年提出的Cayley变换。这一结果已作为研究论文发表。证明了泊松-华核的调和性与定义域的对称性是等价的。此外，即使我们以标准方式改变西格尔域的Koehler度规，泊松-华核是调和的一个充分必要条件是，除了域的对称性外，所考虑的度规是Bergman度规的正数倍。这个结果也已经作为一篇研究论文发表了。我们证明了当且仅当定义管域的凸锥是对称的，解析连续伪逆映射保留了管域。在实际定理中，伪逆映射配备了一个参数，使定理更一般一些。该结果作为研究论文发表（与博士生Chifune Kai共同发表）。我们研究了在上面1中提到的Cayley变换下均匀管域图像的凸性。Cayley变换和对偶Cayley变换的象是凸的充分必要条件是管域是对称的。实际定理是用一种更一般的方式来证明的通过考虑凯莱变换族。该结果作为研究论文发表（与博士生Chifune Kai共同）。我们发现了一系列非对称开凸锥，使得相关的基本相对不变量的度数为1,2，…，r，其中r为开凸锥的秩。此外，在与Ishi的共同研究过程中，我们还证明了每一个基本相对不变量都是复数族右乘法算子的消元的不可约因子。到目前为止，基本的相对不变量只是作为一个简单的算法结果得到的。我们目前正在撰写一篇关于这一结果的联合论文。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