Study on symplectic space and its representation-theoretic structure

Takakura与铃木太郎合作，研究了紧李群余共轭轨道乘积的辛商不变量，得到了如下结果：首先，对于任意单连通紧单李群，我们给出了将上述不变量表示为无穷级数的一般公式。其次，对于三维特殊酉群，我们得到了将不变量表示为有限和的另一个公式。这两个结果都是Takakura关于二维特殊酉群的早期结果的推广。值得注意的是，第一个结果可以看作是二维Yang-Mills理论中Witten体积公式的类比。我们的方法是通过辛商的基本定理，将特征数的计算归结为紧李群或复李代数的表示论问题。更明确地说，我们考虑不可约…的张量积的平凡部分更多的LE表示及其渐近行为。利用Weyl积分公式和仿射李代数的Verlinde恒等式对它们进行分析。另一方面，Miyoshi研究人员得到了曲面丛的Euler类的光滑性的结果。Ochiai研究人员得到了Heun微分方程解的非对易谐振子和联络问题，与有限单群超几何函数有关的多项式(具有M.Yoshida)，关于绝对导数和Zeta函数(具有N.Kurokawa和M.Wakabayashi)，关于某些向量值超几何微分方程解的显式公式(具有M.Fujii)，关于大自由度完全可积系统的分类(与T.Oshima)，关于定位圈的交理论(与K.Mimachi和M.Yoshida)，以及关于Painleve方程的某些多项式解的系数的数论性质(与M.Kaneko)。较少

The head investigator Takakura, in collaboration with Taro Suzuki, studied invariants of symplectic quotients for the products of coadjoint orbits of compact Lie groups, and obtained the following results.First, for any simply-connected compact simple Lie group, we derived a general formula which expresses the above invariant as an infinite series. Secondly, for the 3 dimensional special unitary group, we obtained an another formula which expresses the invariant as a finite sum. Both of them are generalizations of earlier results by Takakura for the 2 dimensional special unitary group. Note that the first result may be regarded as an analogue of Witten's volume formula in 2 dimensional Yang-Mills theory. Our method is to reduce, via the fundamental theorem for symplectic quotients, the computation of characteristic numbers to a problem of representation theory of compact Lie groups or complex Lie algebras. More explicitly, we consider the trivial part of the tensor product of irreducib … More le representations and its asymptotic behavior. We can analyze them by the Weyl integration formula and the Verlinde identity for affine Lie algebras.On the other hand, the investigator Miyoshi obtained a result on the smooth representability of the Euler classes of surface bundles.The investigator Ochiai obtained results on non-commutative harmonic oscillator and the connection problem for the Heun differential equation, on polynomials associated with the hypergeometric functions with finite monodromy groups (with M.Yoshida), on absolute derivations and zeta functions (with N.Kurokawa and M.Wakabayashi), on explicit formulas for solutions of some vector-valued hypergeometric differential equations (with M.Fujii), on classification of completely integrable systems with large degree of freedom (with T.Oshima), on intersection theory for located cycles (with K.Mimachi and M.Yoshida), and on number-theoretic property for coefficients of certain polynomial solutions of the Painleve equations (with M.Kaneko). Less