Study on geometry and various invariants of symplectic space

首席调查员高仓健（与T。Suzuki）发表了一个关于三度特殊酉群的几个余伴轨道的乘积的辛商的体积的显式公式的结果。体积被看作是交对的生成函数，与不可约表示张量积的不变子空间的渐近维数一致。我们还考虑了任何紧李群的类似问题，并推导出一个公式，表示渐近维数作为一个有限和。这是对特殊的低次酉群的推广。然而，在这种推广中，我们只处理所有最高权重都是正则的情况。其他案件仍有待研究。此外，由于每一项都是作为超几何积分给出的，因此公式不是完全明确的。研究这些积分是很有趣的。另一方面，几年前，我们得到了另一个公式，它将渐近维数表示为无穷级数。然而，事实证明，推导有一个微妙的困难，涉及双重极限过程。因此，我们在所有最高权都是正则的假设下，给出了这个无穷级数公式的严格证明。研究员落合获得的结果，不变分布的伪黎曼对称空间，频谱zeta函数的非交换谐振子，线性关系的多zeta值（与K。Ihara），双曲3-锥流形上的调和向量场（M. Fujii），关于伪射影平面的算术结构（与F. Kato），多重Gamma函数和Mahler测度（N. Kurokawa），关于对称对的幂零轨道（K. Nishiyama和C. B。Zhu），以及一类微分方程的逐次逼近（T.服部）。

The head investigator Takakura (with T. Suzuki) published the result on an explicit formula for the volume of a symplectic quotient for a product of several coadjoint orbits of the special unitary group of degree three. The volume is regarded as a generating function of intersection pairings, and coincides with the asymptotic dimension of the invariant subspace of tensor product of irreducible representations. We also considered the similar problem for any compact Lie group, and derived a formula that expresses the asymptotic dimension as a finite sum. This is a vast generalization of those for the special unitary group of low degrees. However, in this generalization, we only treat the case when all the highest weights are regular. The other cases remain to be studied. In addition, since each term is given as a hypergeometric integral, the formula is not completely explicit. It is interesting to investigate these integrals. On the other hand, several years ago, we obtained another formula which expresses the asymptotic dimension as an infinite series. However, it turned out that the derivation has a subtle difficulty, concerned with the double limit process. So, we gave a rigorous proof of this infinite series formula, under the assumption that all highest weights are regular.The investigator Miyoshi obtained a result on spinable foliations. The investigator Ochiai obtained results, on invariant distributions on pseude-Riemannian symmetric spaces, on spectral zeta function of non-commutative harmonic oscillators, on linear relations for multiple zeta values (with K. Ihara), on harmonic vector fields on hyperbolic 3-cone-manifold (with M. Fujii), on arithmetic structure of fake projective planes (with F. Kato), on multiple gamma functions and Mahler measures (with N. Kurokawa), on nilpotent orbits for symmetric pairs (with K. Nishiyama and C. B. Zhu), and successive approximations for a certain differential equation (with T. Hattori).