Functional equations of spherical functions on p-adic homogeneous spaces

LetG be a connected reductive linear algebraic group andX aG-homogeneous affine algebraic variety both defined over a p-adic field k, where we assume a minimalk-parabolic subgroup ofG acts with open orbit. We are interested in spherical functions onX =X(k). In the present papaer, we give a unified method to obtain functional equations of spherical functions on X under the condition (AF) in the introduction, and explain functional equations are reduced to those ofp-adic local zeta functions of small prehomogeneous vector spaces of limited type.