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 $\varphi$ on a $p$-adic group $G$ we write down an explicit linear combination ${\varphi }^H$ of Deligne-Lusztig functions on an endoscopic group $H$ such that $\varphi$ and ${\varphi }^H$ have ``matching orbital integrals’'. In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on $p$.