Representation and analysis on symmetric spaces

首先，我们给出了紧李群关于两个对称子群的双陪集分解。这是Oshima-Sekiguchi和Hoogenboom的结果的统一推广，没有假设两个对合的交换性。利用几个典型的球面子群的作用，计算了旗流形上的轨道结构。但是我们还没有得到缠绕函数（“广义球函数”）的精确结构。其次，我们证明了对称群的Hecke代数是由量子一般线性群自然产生的。这一结果被应用于构造Specht模。第三，确定了Cartan型李代数在其自然表示的m重张量积中的交换代数。在本文中，我们找到了经典Schur对偶的一个类比，最后给出了与对称矩阵和Hermitian矩阵相关联的轨道p-adic zeta函数的一个显式公式，并研究了四维流形上的奇异自由作用。

First, we descibed double coset decompositions of compact Lie groups with respect to two symmetric subgroups. This is a unified generalization of the results given by Oshima-Sekiguchi and Hoogenboom, without assuming commutativity of two involutions. We also computed orbit structure on flag manifolds by the action of som typical spherical subgroups. But we have not yet obtained precise structure of intertwining functions ("generalized spherical functions").Secondly, we showed that Hecke algebras of the symmetric groups arise naturally from quantum general linear groups. This result is applied to consruct Specht modules. We also studied p-adic spherical homogeneous spaces and their spherical functions.Thirdly, we determined the commutant algebra of a Cartan type Lie algebra in the m-fold tensor product of its natural representations. In this study, we found an analogy of the classical Schur's duality.Lastly, we gave an explicit formula of orbital p-adic zeta functions associated to symmetric and hermitian matrices.We also studied exotic free actions on 4-dimensional manifolds.