Spherical functions on spherical varieties

Let X = H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K = G(o) a hyperspecial maximal compact subgroup of G= G(0). We compute eigenfunctions ("spherical functions") on X = X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicity-free case (namely, that an H-invariant functional on an irreducible unramified representation π is non-zero on π^K), (2) the unramified Plancherel formula for X, including a formula for the "Tamagawa measure" of X(o), and (3) a computation of the most continuous part of H-period integrals of principal Eisenstein series.