On endoscopic transfer of Deligne-Lusztig functions

TL;DR

This paper proves the fundamental lemma for Deligne-Lusztig functions, establishing explicit matching of orbital integrals between a p-adic group and its endoscopic group, confirming a conjecture of Kottwitz.

Contribution

It provides an explicit construction of matching functions for Deligne-Lusztig functions on p-adic groups and their endoscopic counterparts, confirming Kottwitz's conjecture under mild restrictions.

Findings

Established the fundamental lemma for Deligne-Lusztig functions.

Constructed explicit matching functions between groups.

Confirmed Kottwitz's conjecture under certain conditions.

Abstract

In this paper we prove the fundamental lemma for Deligne-Lusztig functions. Namely, for every Deligne-Lusztig function $\phi$ on a $p$-adic group $G$ we write down an explicit linear combination $\phi^H$ of Deligne-Lusztig functions on an endoscopic group $H$ such that $\phi$ and $\phi^H$ have ``matching orbital integrals''. In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on $p$.