Spherical functions on p-adic homogeneous spaces.

设G是定义在p进域k上的代数群，X是G和X = X (k)的齐次空间。设H（G，K）是G = G(K)关于G的极大紧子群K的Hecke代数。X上的H（G，K）公特征函数称为球函数，它是数论和表示论中都感兴趣的对象。在X上的抛物子群轨道上的一定条件下，利用泛函方程和群的球函数给出了球函数的显式公式。我们可以把它应用到对称形式和厄米特形式的空间中，它们是数论中有趣的对象。在这些空间中，球面函数可以被看作是局部表示密度的生成函数，因此它们是密切相关的。得到对称形式和厄米特形式表示的局部密度的显式公式是数论中的一个经典问题。对于非分支厄米形式的空间X，我们用上述方法确定了球面函数的显式公式，并将schwarz - bruhat函数的空间结构S（K\X）作为Hecke模。我们还利用x上的球函数得到了两类局部密度的显式公式。通过与F. Sato的共同研究，我们得到了p≠2，一般尺寸下对称形式的局部密度的完整显式公式。该方法是利用Iwahori子群的作用对对称形式进行分类，并计算一定的高斯和。将上述方法推广到厄米特形式的情况下，我们得到了一般尺寸下p≠2厄米特形式的局部密度的显式公式。

Let G be an algebraic group defined over a p-adic field k, X a homogeneous space of G and X = X (k).Let H(G,K) be the Hecke algebra of G = G(k) with respect to a maximal compact subgroup K of G.An H(G,K)-common eigenfunction on X is called a spherical function, which is an intereted object both in Number Theory and Representation Theory.Under certain condition on orbits by a parabolic subgroup in X, we have given a method to obtain explicit formulas of spherical functions by means of functional equations and spherical functions of groups. We can apply it to the spaces of symmetric forms and hermitian forms, which are interesting objects in Number theory. In these spaces, spherical functions can be viewed as generating functions of local densities of representation, so they are closely related. It is a classical problem in Number Theory to obtain explicit formulas of local densities of representations for symmetric forms and hermitian forms.For the space X of unramified hermitian forms, we have determined an explicit formula for spherical functions by the above method and the structure of the space of Schwartz-Bruhat functions S(K＼X) as Hecke module. We have also obtained two kinds of explicit formulas of local densities by using spherical functions on X.By a joint research with F. Sato, we have obtained a complete explicit formula of local densities of symmetric forms for p ≠ 2, in general sizes. The method is to classify symmetric forms by the action of Iwahori subgroup and calculate certain Gaussian sums.Extending the above method to the the case of hermitina forms, we have obtained an explicit formula of local densities of hermitian forms for p ≠ 2, in general sizes.