Arthur packets and the Ramanujan conjecture

Abstract

The purpose of this paper is to show that under a part of generalized Arthur's A-packet conjecture, locally generic cuspidal automorphic representations of a quasisplit group over a number field are of Ramanujan type, i.e., are tempered at almost all places. The A-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands L-packet which is then answered here in its generality.

On equality of arithmetic and analytic factors through local Langlands correspondence

“…A result like this has an interesting application in proving the generic A-packet conjecture discussed in [Shahidi 2011]. This is a kind of converse to the tempered L-packet conjecture, which asserts that every tempered L-packet of a quasisplit group has a generic member [Shahidi 1990;Vogan 1978].…”

“…Denote by L G its L-group. Let W F be the Weil-Deligne group of F. Let ρ : W F → L G be an admissible homomorphism (see [Arthur 1984;Shahidi 2011]). Let r be an irreducible complex representation of L G on a finite dimensional complex vector space V , i.e., r : L G → GL(V ) is an analytic homomorphism.…”

“…We note that the elements of φ ψ are supposed to provide the main nontempered members of ψ (see [Arthur 1984]), i.e., those which have not already appeared in other A-packets. The proof given in [Shahidi 2011] is based on the matching of only L-functions for certain Levi factors through LLC.…”

“…Moreover, the examples of 3 discussed below should handle some cases of exceptional groups. More precisely, using [Shahidi 2011] the work in [Kim 2012] proves that if ψ is an Arthur packet for GSpin(F), where F is a p-adic field, then the Langlands packet φ ψ attached to ψ has a generic member only if φ ψ is tempered. This clearly gives a converse to the tempered (or generic) L-packet conjecture [Shahidi 1990;Vogan 1978].…”

“…They include the cases of twisted exterior square L-functions for GL(n) as well as twisted exterior cube for GL(6). This equality can be used to prove special cases of the generic Arthur packet conjecture [Arthur 1984;Shahidi 2011] as we explain in Section 3. Finally in Section 4 we address the issue of stability of γ -factors within our method and discuss the progress made on it and some of its consequences.…”