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.