Study on geometry and various invariants of symplectic space

首席研究员Takakura(与T.Suzuki)发表了关于三次特殊酉群的几个余共轭轨道的乘积的辛商体积的显式公式的结果。体积被认为是交对的母函数，并且与不可约表示的张量积的不变子空间的渐近维度重合。对于任何紧李群，我们也考虑了类似的问题，并导出了一个将渐近维数表示为有限和的公式。这是对低度特殊酉群的推广。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。其他案例仍有待研究。此外，由于每一项都以超几何积分的形式给出，所以公式不是完全显式的。研究这些积分是很有趣的。另一方面，几年前，我们得到了用无穷级数表示渐近维的另一个公式。然而，事实证明，这个推导有一个微妙的困难，涉及到双重限制过程。因此，在所有最高权都是正则的假设下，我们给出了这个无穷级数公式的严格证明。调查者Miyoshi得到了关于可旋叶的一个结果。Ochiai获得了伪黎曼对称空间上的不变分布、非对易谐振子的谱Zeta函数、多个Zeta值的线性关系(与K.Ihara)、双曲3锥流形上的调和向量场(与M.Fujii)、伪射影平面的算术结构(与F.Kato)、多重Gamma函数和Mahler测度(与N.Kurokawa)、对称对的幂零轨道(与K.Nishiyama和C.B.朱)以及一类微分方程的逐次逼近(与T.Hattori)的结果。

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).