Abstract: First I will present a geometric method, due to Kazhdan, of approximating representation theory of reductive groups over local fields of positive characteristic (like the field F_p((t)) of Laurent power series with coefficients in a finite field) with representation theory of reductive groups over local fields of zero characteristic (like the field Q_p of p-adic numbers).
Then I will present a generalization of this method, due to Gourevitch, Avni and myself, which approximates harmonic analysis over spherical varieties over local fields of positive characteristic with harmonic analysis over spherical varieties over local fields of zero characteristic.
As an application we show that (GL(n+1,F),GL(n,F)) is a strong Gelfand pair for all local fields F of positive characteristic. This means that the restriction to GL(n,F) of every irreducible smooth representation of GL(n+1,F) "decomposes" with multiplicity one. We use our method to
deduce this from the zero characteristic case, which was proven in 2007 by Gourevitch, Rallis, Schiffmann and myself.