Asai L-functions and Jacquet's conjecture

Abstract

We give a conceptually simple proof of the square-integrable case of a conjecture of Jacquet concerning distinguished representations of the general linear group over a local field of characteristic zero. The proof is based on consideration of the Rankin-Selberg integral representation of the generalized Asai L-function and utilizes global methods.

Distinguished representations and poles of twisted tensor 𝐿-functions

“…established in Theorem 6 of [11]. One has only to note that the Rankin-Selberg L-function on the left can have at most a simple pole at s = 0 when π is in the discrete series.…”

“…The first and third named authors initially had a preprint that contained results of the flavour of Theorems 1.1 and 1.4. Then they came to know about a paper of the second named author in which a study of the poles of the twisted tensor L-function is carried out, resulting in a proof of the square integrable case of Jacquet's conjecture about distinguished representations [11]. Around that time, the second named author was also thinking about a statement like Corollary 1.6, which refines Jacquet's conjecture in this case.…”

“…There has been a lot of interest in the study of these representations and their global counterparts ever since the work of Harder, Langlands and Rapoport [10]. For various aspects of the local theory, we refer to [1], [6], [9], [11], [12], [13], and the references therein.…”

Distinction of some induced representations

“…From the proof of Theorem 1 of [Kab04], it follows that out of the hyperplanes in (u, s) defined by c ρ u is the representation of G n−j (F ) called the j-th derivative of ρ u (see summary before Proposition 2.3 of [UAR04]), for j from 1 to n, the space of solutions of equation 1 is of dimension at most one. If we take a basis of (f α ) α∈A of F ρ , the polynomial family over the irreducible complex…”

“…According to Proposition 26 in [Fli88], Proposition 12 of [Fli91], Theorem 6 of [Kab04], and Corollary 1.6 [UAR04], our result reduces the proof of the conjecture to show that representations of the form ∆ 1 × · · · × ∆ t with ∆ σ i+1 = ∆ ∨ i for i = 1, 3, .., 2r − 1 for some r between 1 and t/2, and non isomorphic distinguished or η-distinguished ∆ i 's for i > 2r are not distinguished whenever one of the ∆ i 's is η-distinguished for i > 2r. According to [Mat09a], the preceding conjecture implies the equality of the analytically defined Asai L-function and the Galois Asai L-function of a generic representation.…”