On the endoscopic classification of representations of quasi-split classical groups

The endoscopic classification of representations of quasi-split symplectic and orthogonal groups is a celebrated result of Arthur (extended to quasi-split unitary groups by Mok) which has had wide applications to representation theory and number theory. It is a collection of many interrelated statements that in particular gives a classification of the irreducible admissible representations of such groups over local fields, and the discrete automorphic representations of such groups over number fields, in terms of A-packets.

Until recently this result was conditional on a number of unproven statements, in particular the construction, character identities, and intertwining relations, of co-tempered A-packets over non-archimedean local fields. These statements have now been proved in joint work with Atobe,Gan, Ichino, Minguez, and Shin, rendering Arthur’s result conditional only on the validity of the weighted fundamental lemma, which is independently being completed.

In this talk I will explain the general statements in Arthur’s classification, the role played by co-tempered A-packets, mention some applications, and touch on the work that supplied the missing results.