In this paper we prove the fundamental lemma for Deligne–Lusztig functions. Namely, for every Deligne–Lusztig function on a -adic group we write down an explicit linear combination of Deligne–Lusztig functions on an endoscopic group such that and have “matching orbital integrals.” In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on .
"On endoscopic transfer of Deligne–Lusztig functions." Duke Math. J. 161 (4) 675 - 732, 15 March 2012. https://doi.org/10.1215/00127094-1548371