Almost 5 years ago (time flies), I asked in Rankin-Selberg convolution and product of degrees whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ and
$\operatorname{GL}_{n'}(\mathbb{A}_{\mathbb{Q}})$ gave rise to an automorphic representation of $\operatorname{GL}_{n.n'}(\mathbb{A}_{\mathbb{Q}})$. Paul Garrett answered it by giving the known cases where this was proven at that time.

Have there been breakthroughs so far getting us any closer to such a general result?

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    $\begingroup$ I don't think so. It's a hard problem. $\endgroup$ – Kimball Dec 25 '19 at 15:06

Newton and Thorne proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ corresponding with a holomorphic cuspidal newform of even integral weight $k\geq 2$, squarefree level, and trivial central character, then for each $n\geq 1$, the $n$-th symmetric power lift $\mathrm{Sym}^n \pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{n+1}(\mathbb{A}_{\mathbb{Q}})$. We have the standard identity

$\mathrm{Sym}^n \pi\otimes\mathrm{Sym}^n\pi = \boxplus_{j=0}^n \mathrm{Sym}^{2j}\pi$,

so by Newton--Thorne, this is an isobaric automorphic representation of $\mathrm{GL}_{(n+1)^2}(\mathbb{A}_{\mathbb{Q}})$. This is not yet known to hold for $\pi$ corresponding with Hecke--Maass forms. Thus this is a thin set of examples, but I find it noteworthy nonetheless. This showed up on the arxiv the day before you posted your question.

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  • $\begingroup$ Thank you very much for your answer. Are there known important properties, even conjectural, that are stable under RS convolution ? $\endgroup$ – Sylvain JULIEN Mar 4 at 18:28

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