On endoscopic transfer of Deligne–Lusztig functions

Abstract

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 $p$.