Thomas Haines

Abstract: The Langlands-Shelstad theory of endoscopy plays a central role in the study of Shimura varieties and the Arthur-Selberg trace formula. The fundamental lemma and a deep consequence, endoscopic transfer, have now been established in works of Ngo, Waldspurger, and Hales. Both of these statements are identities involving orbital integrals of a function $f$ on a $p$-adic group $G$ and those of a certain "transfer" function $f^H$ on a related group $H$, called endoscopic. This talk will give background on the fundamental lemma, some of its variants, and the roles they played. Then I will describe a conjectural construction of a large class of matching functions in the Bernstein centers of $G$ and $H$. This conjecture has been verified in some encouraging cases. The functions being transferred arise in connection with Shimura varieties with bad reduction, and some applications to that subject will also be briefly discussed.