How strong is the requirement of being a Gelbart-Jacquet lift?

Let $$\pi$$ be a cuspidal automorphic representation of $$\mathrm{GL}(3)$$ over a number field $$F$$. I am wondering how general are Gelbart-Jacquet lifts of automorphic representations of $$\mathrm{GL}(2)$$ among these.

I have a very limited understanding of Gelbart-Jacquet lifts: they are the cuspidal automorphic representations of $$\mathrm{GL}(3)$$ such that $$L(s, \mathrm{sym}^2 \pi)$$ has a pole at $$s=1$$. I would like to precisely understand what they are as automorphic objects, in order to understand how strong is this hypothesis and how it can be used in practice to appeal to the $$\mathrm{GL}(2)$$ setting. Any precise reference for it would be welcome.

The Gelbart-Jacquet lift of a $$\mathrm{GL}_2$$ cuspidal automorphic representation $$\pi$$ is an automorphic representation $$\Pi$$ of $$\mathrm{GL}_3$$. More generally, the Gelbart-Jacquet lift $$\Pi$$ is a $$\mathrm{GL}_3$$ automorphic representation associated to a $$\mathrm{GL}_2$$ automorphic representation $$\pi$$ and a Hecke character $$\chi$$ such that $$L(s,\pi \otimes \widetilde{\pi} \otimes \chi) = L(s,\Pi) L(s, \chi)$$. one can use the $$\mathrm{GL}_3$$ converse theorem to show that $$L(s,\Pi)$$ is an $$L$$-function of an automorphic representation of $$\mathrm{GL}_3$$.

When $$\chi = 1$$, $$\Pi$$ is equal to $$\mathrm{ad} \pi$$, the adjoint. When $$\chi = \omega_{\pi}^{-1}$$, the inverse of the central character of $$\pi$$, so that $$\widetilde{\pi} \otimes \chi = \pi$$, $$\Pi$$ is equal to $$\mathrm{sym}^2 \pi$$. (In particular, these two are equal up to a twist.)

As an aside, note that the Gelbart-Jacquet lift need not be cuspidal; see Poles of $L$-functions associated to Maass forms.

Gelbart-Jacquet lifts are rare among all $$\mathrm{GL}_3$$ automorphic representations.

The first way to quantify this is a result of Ramakrishnan; which states that there is (essentially) a bijection (after twisting, if necessary) between Gelbart-Jacquet lifts and self-dual automorphic representations of $$\mathrm{GL}_3$$. Unlike for $$\mathrm{GL}_2$$, being self-dual is a very restrictive statement for $$\mathrm{GL}_n$$ with $$n \geq 3$$.

The second way to quantify this is via the Weyl law. A result of Guerreiro gives a Weyl law for Gelbart-Jacquet lifts with main term $$\asymp T^4/\sqrt{\log T}$$, whereas the Weyl law for all $$\mathrm{GL}_3$$ forms has main term $$\asymp T^5$$.

A third way of viewing this is to look locally. Let $$\Pi$$ be an automorphic representation of $$\mathrm{GL}_3$$. At almost all places $$v$$, the local component $$\Pi_v$$ is an unramified principal series representation $$|\cdot|_v^{t_{v,1}} \boxplus |\cdot|_v^{t_{v,2}} \boxplus |\cdot|_v^{t_{v,3}}$$, where the spectral parameters $$t_{v,1},t_{v,2},t_{v,3}$$ satisfy $$t_{v,1} + t_{v,2} + t_{v,3} = 0$$ and $$|\Re(t_{v,j})| < 1/2$$ (the best known bound towards the Ramanujan conjecture in this case is $$5/14$$ due to Blomer-Brumley). Self-dual representations satisfy the additional (quite stringent!) restriction that $$t_{v,j} = 0$$ for some $$j \in \{1,2,3\}$$.

• If I understand well, the conjunction of self-duality and Ramanujan conjecture implies the set $\{t_{v,j}\}$ is invariant under the action of the complex conjugation right? Is there any link with RH? – Sylvain JULIEN Apr 29 '19 at 21:12
• Also does the definition of the adjoint L-function imply $\pi$ and $\tilde{\pi}$ are orthogonal in some sense? – Sylvain JULIEN Apr 29 '19 at 21:18
• @SylvainJULIEN yes, no, and no to your three questions. – Peter Humphries Apr 29 '19 at 21:19
• Thank you for the fast answer. – Sylvain JULIEN Apr 29 '19 at 21:21
• Thanks Peter, this is a great and complete answer. – Desiderius Severus May 20 '19 at 2:34