Local and global questions "beyond endoscopy"

Abstract

The near-completion of the program of endoscopy poses the question of what lies next. This article takes a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R. P. Langlands, aiming to extract from the stable trace formula of a group G the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of L-groups ${^LH} \to {^LG}$. With the extension of the problem of functoriality to the "relative" setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman--Kazhdan, also around 2000, to generalize the Godement--Jacquet proof of the functional equation to arbitrary L-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.