Representation and analysis on symmetric spaces

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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.