On Endoscopic Transfer of Deligne–lusztig Functions

In this paper we prove the fundamental lemma for Deligne–Lusztig functions. Namely, for every Deligne–Lusztig function φ on a p-adic group G we write down an explicit linear combination φ of Deligne–Lusztig functions on an endoscopic group H such that φ and φ have “matching orbital integrals”. In particular, we prove a conjecture of Kottwitz [Ko4]. More precisely, we do it under some mild restriction on p.