Nearby cycles of parahoric shtukas, and a fundamental lemma for base change

Abstract

Using the Langlands–Kottwitz paradigm, we compute the trace of Frobenius composed with Hecke operators on the cohomology of nearby cycles, at places of parahoric reduction, of perverse sheaves on certain moduli stacks of shtukas. Following an argument of Ngô, we then use this to give a geometric proof of a base change fundamental lemma for parahoric Hecke algebras for GL[subscript n] over local function fields. This generalizes a theorem of Ngô, who proved the base change fundamental lemma for spherical Hecke algebras for GL[subscript n] over local function fields, and extends to positive characteristic (for GL[subscript n]) a fundamental lemma originally introduced and proved by Haines for p-adic local fields.