Spherical functions on p-adic homogeneous spaces.

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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.