On the geometric Ramanujan conjecture

Abstract

In this paper we prove two results pertaining to the (unramified and global) geometric Langlands program. The first result is an analogue of the Ramanujan conjecture: any cuspidal D-module on Bun_G is tempered. We actually prove a more general statement: any D-module that is *-extended from a quasi-compact open substack of Bun_G is tempered. Then the assertion about cuspidal objects is an immediate consequence of a theorem of Drinfeld-Gaitsgory. Building up on this, we prove our second main result, the automorphic gluing theorem for the group SL_2: it states that any D-module on Bun_{SL_2} is determined by its tempered part and its constant term. This theorem (vaguely speaking, an analogue of Langlands' classification for the group SL_2(R)) corresponds under geometric Langlands to the spectral gluing theorem of Arinkin-Gaitsgory and the author.