Analytic and geometric studies of group representations and their applications

首席研究员野村，研究代数系统的泛函分析方法，研究了黎曼希尔伯特流形的幂等元（格拉斯曼流形）的希尔伯特约旦代数和描述测地线，黎曼度量和截面曲率公式，通过使用约旦代数结构，阐明了贡献的代数。研究了与无重数紧李群作用相关的Berezin变换的谱分解。其中一些结果已经在鸟取的工作室和藤田的硕士论文（京都大学）中报道。研究员梅田研究了不变量理论和表示理论。在“基于量子对称性的不变量理论”的主题下，一个特别的焦点被放在与对偶对相关的Capelli恒等式的研究上。得到了对偶对O_n，{/gothsp}_{Zm}$的Capelli恒等式的一个公式.这解释了一些与多重性无关的作用相关的卡佩利恒等式，也解释了${/goth o}_-n}$的通用包络代数的中心元素。虽然不是一个最终的完整形式，结果报告在G. Ochial的硕士论文（京都大学）。研究员Hiral已经处理了无限对称群和流形的同构群。考虑到自然表示的无限张量积，人们发现某个无限对称群充当交织算子。证明了这两类群在一定情况下构成一个对偶对，并研究了这个无限张量积通过对偶对的不可约分解。

The Head Investigator Nomura, studying algebraic systems by functional-analytic method, has investigated the Riemann-Hilbert manifold of idempotents (Grassmann manifolds) of Hilbert Jordan algebras and described geodesics, the Riemannian metric and the sectional curvature formula by using the Jordan-algebra structure, clarifying the contribution of the algebra. Spectral decomposition of Berezin transformation associated to the multiplicity-free compact Lie group actions has also been studied. Some of the results are already reported at the work-shop in Tottori and in E.Fujita's master thesis (Kyoto university).Investigator Umeda has studied the invariant theory and the representation theory. A particular focus has been placed upon the research on the the Capelli identity associated to dual pairs under the theme "invariant theory based on quantum symmetry". A formula for the Capelli identity associated to the dual pair $O_-n, {/goth sp} _- {Zm} $ has been obtained. This explains some of the Capelli identities associated to multiplicity-free actions and also the central elements of the universal enveloping algebra of $ {/goth o} _-n} $. Though not an ultimately complete form, the results are reported in G.Ochial's master thesis (Kyoto University).Investigator Hiral has treated the infinite symmetric groups and the diffeomorphism group of manifolds. Considering the infinite tensor product of natural representations, one finds that a certain infinite symmetric group acts as intertwining operators. It is shown that these two kinds of groups form a dual pair in a certain case and investigation has been done on the irreducible decomposition of this infinite tensor product through the dual pair.