Spherical pairs over close local fields

Abstract

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 (\mathrm{GL}_{n+1}(F),\mathrm{GL}_n(F)) is a strong Gelfand pair for all local fields of arbitrary characteristic, and that the pair (\mathrm{GL}_{n+k}(F),\mathrm{GL}_n(F)\times\mathrm{GL}_k(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.