Spherical Pairs Over Close Local Fields

TL;DR

This paper extends harmonic analysis results from characteristic zero fields to positive characteristic fields, establishing Gelfand pair properties for certain pairs of general linear groups over local fields.

Contribution

It generalizes Kazhdan's results to positive characteristic fields and proves new Gelfand pair properties for (GL_{n+1},GL_n) and (GL_{n+k},GL_n x GL_k).

Findings

(GL_{n+1},GL_n) is a strong Gelfand pair over all local fields.

(GL_{n+k},GL_n x GL_k) is a Gelfand pair over fields of odd characteristic.

Provides a condition for finite generation of modules over the Hecke algebra.

Abstract

Extending results of Kazhdan 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. One of the Ingredients of the proof is a condition for finite generation of some modules over the Hecke algebra. We apply our results to show that the pair (GL_{n+1},GL_n) is a strong Gelfand pair for all local fields, and that the pair (GL_{n+k},GL_n x GL_k) is a Gelfand pair for all local fields of odd characteristic.