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

加藤与高野共同研究了p进场上对称空间上的球函数。利用极大紧子群下对称空间的轨道分解（Cartan分解，其一般公式仍为猜想形式），得到了球面函数在环面上的一个麦克唐纳型公式，该公式用对称空间的Weyl群（小Weyl群）的和来表示其值。一般来说，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换在内的几个例子，我们有了这样的公式。作为对称空间研究的副产品，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，是关于约化群的对称子群的区分可容许表示）：我们成功地建立了Jacquet子表象的一个相对版本（=对称空间版本）. More归一定理，该定理断言对于P进约群G的任何不可约容许表象V，存在至少一个抛物形P和一个不可约倒转形W，使得在Cartan分解的假设下，V可以嵌入到P的诱导表象W中。即，通过定义相对可约性的概念，我们证明了对称空间的任何不可约表示都可以嵌入到由sigma分裂抛物子群及其Levi子群的不可约区分表示组成的对所关联的诱导表示中。这个结果可以看作是将p进群上的调和分析推广到对称空间上的调和分析的第一步。利用“相对倾向性”的概念在p进群和/或其他群上建立对称空间的表示理论是一个有趣的问题。其他研究者也在自同构表示和自同构形式（H.Saito和A.Murase）以及实李群的结构理论和表示理论（T.Matsuki和K.Nishiyama）方面取得了一些成果。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