# Rankin-Selberg convolution and product of degrees as of Christmas 2019

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?

• I don't think so. It's a hard problem. – 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.