Functional equations of spherical functions on p-adic homogeneous spaces

Let dubble struck G sign be a connected reductive linear algebraic group and dubble struck X sign a dubble struck G sign-homogeneous affine algebraic variety both defined over a p-adic field k, where we assume a minimal k-parabolic subgroup of dubble struck G sign acts with open orbit. We are interested in spherical functions on X = 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 of p-adic local zeta functions of small prehomogeneous vector spaces of limited type.