Spherical pairs over close local fields

Extending results of [Kaz86] to the relative case, we relate harmonic analysis over some spherical spaces G.(F)/H.(F), where F is a field of positive characteristic, to harmonic analysis over the spherical spaces G.(E)=H.(E), where E is a suitably chosen field of characteristic 0. We apply our results to show that the pair .(GLnC1.F ); GL _n.(F)) is a strong Gelfand pair for all local fields of arbitrary characteristic, and that the pair .GL _nCk.(F); GL _n.(F)̃GLk.(F)) is a Gelfand pair for local fields of any characteristic different from 2. We also give a criterion for finite generation of the space of K-invariant compactly supported functions on G.E/=H.E/ as a module over the Hecke algebra.