# Spherical pairs over close local fields

### Avraham Aizenbud

The Weizmann Institute of Science, Rehovot, Israel### Nir Avni

Harvard University, Cambridge, USA### Dmitry Gourevitch

Institute for Advanced Study, Princeton, USA

## 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.

## Cite this article

Avraham Aizenbud, Nir Avni, Dmitry Gourevitch, Spherical pairs over close local fields. Comment. Math. Helv. 87 (2012), no. 4, pp. 929–962

DOI 10.4171/CMH/274