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 ^H of Deligne-Lusztig functions on an endoscopic group H such that φ and φ ^H have "matching orbital integrals." In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on p.