Representation theoretic research of spherical functions on p-adic homogeneous spaces

S.Kato与K.Takano一起研究了p-ady域上对称空间上的球函数。利用对称空间在极大紧子群下的轨道分解(Cartan分解，其一般公式仍然是猜想形式)，得到了球函数的Macdonald型公式，它用对称空间的Weyl群(小Weyl群)上的和来表示环面上的值.关于球函数的显式公式的问题仍然存在。然而，对于包括辛群的二次基变化在内的几个例子，我们有这样的公式。作为对对称空间研究的一个副产品，我们得到了关于对称空间表示的一个表示理论结果(更确切地说，关于约化群对称子群的可区别允许表示)：我们成功地建立了Jacquet子表示…的一个相对版本(=对称空间版本更多的刻划定理，它断言对于p-adi约化群G的任一不可约可容许表示V，至少存在一个抛物线P和一个不可约尖点W，使得在Cartan分解的假设下，V可以嵌入到由P到W的诱导表示中。也就是说，通过定义相对尖点性的概念，我们证明了对称空间的任何不可约表示都可以嵌入到一个诱导表示中，该诱导表示与一个由sigma-分裂抛物子群和它的Levi子群的一个不可约可区别表示组成的对有关。这一结果可以看作是将p-进群调和分析推广到对称空间上的第一步。利用“相对凹凸性”的概念，在不同领域的p-ady群和/或其他群上建立对称空间的表示理论是很有趣的。其他研究者也得到了关于自同构表示和自同构形式的一些结果(H.Saito和A.Murase)，以及关于实李群的结构理论和表示理论(T.Matsuki和K.Nishiyama)。较少

S.Kato, the Head investigator, studied the spherical functions on symmetric spaces over p-adic fields, together with K.Takano. By using orbit decomposition of symmetric spaces under maximal compact subgroups (Cartan decomposition, general formula of which is still in conjectural form), we obtained a Macdonald-type formula for spherical functions which expressed the value on tori by a sum over the Weyl groups of symmetric spaces (the little Weyl groups). The problem to have explicit formulas for spherical functions in general remained. However, for several examples including quadratic base change of symplectic groups, we had such formulas. As a byproduct of our study of symmetric spaces, we obtained a representation theoretical result about the representations of symmetric spaces (more precisely, about distinguished admissible representations for symmetric subgroups of reductive groups) : We succeeded in establishing a relative version (=symmetric space version) of Jacquet's subrepresen … More tation theorem which asserts that for any irreducible admissible representation V of a p-adic reductive group G, there exists at least one parabolic P and one irreducible cuspidal W such that V may be embedded into the induced representation of W from P under the assumption of the Cartan decomposions. Namely, by defining the notion of relative cuspidality, we showed that any irreducible representation of a symmetric space can be embedded in a induced representation associated with a pair consisting of a sigma-split parabolic subgroup and an irreducible distinguished representation of its Levi subgroup. This result can be viewed as a first step to generalize the harmonic analysis on p-adic groups to that on symmetric spaces. It is interesting to build representation theory of symmetric spaces on p-adic groups and/or other groups over various fields by using the notion of "relative cuspidality".Other investigators also obtained several results on automorphic representations and automorphic forms (H.Saito and A.Murase ), and on structure theory and representation theory of real Lie groups (T.Matsuki and K.Nishiyama). Less