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 quasicompact 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.

Automorphic Gluing

“…The proof for the counit β L G ○β G → id follows by combining the Deligne-Lusztig duality of the second author (see [Che20a]) with the Ramanujan conjecture of the first author (see [Ber21b]).…”

“…It turns out that DMod(Bun G ) contains a remarkable full subcategory: the subcategory DMod(Bun G ) temp of tempered objects. Tempered objects were introduced in [AG15, Section 12] and studied in [Ber21c], [Ber21b], [FR21].…”

“…The general definition of DMod(Bun G ) temp will be given later: there are several different ways to present it, but they all involve the Hecke action as a common ingredient. The case of G of semisimple rank one is an exception: for instance, an object F ∈ DMod(Bun SL 2 ) is tempered iff its cohomology with compact supports vanishes, see [Ber21b]. The "only if" direction is true for general G by [Ber21c].…”

The Arinkin-Gaitsgory temperedness conjecture

“…To avoid the subtleties involved in the above argument, one could also proceed as follows. First, by [Ber5] Theorem 1.4.8, KerpAT x ‹ ´q " KerpWS 0 ‹ ´q for WS 0 as in loc. cit., i.e., one takes the unit spherical Whittaker sheaf in Whit sph…”

“…For our point x, let H sph x denote the associated (derived) spherical Hecke category. There is a certain object AT x P H sph x , which we call the anti-tempered unit following [Ber5]. By definition, DpBun G q x-temp is the kernel of the corresponding Hecke functor:…”