If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, independent of $n$) lower and upper bounds for the norm of elements of $S$?


Motivation: denoting by $\mathcal{M}\_{a,b}$ the set of such automorphic representations whose set $S\_{a,b}$ of Satake parameters at unramified primes $p$ fulfill $a\leq\vert\alpha\_{p,j}\vert\leq b$, a proof that all the relevant Satake parameters are bounded above by a quantity independent of $n$ (and $p$ as well) would imply that if $\mathcal{M}\_{a,b}$ is closed under the Rankin-Selberg convolution, then $a=b=1$.



Do we know absolute bounds for the norm of Satake parameters?
We don't know any upper bound for $|\alpha\_{p,j}|$ that is independent of $p$. On the other hand, we do know that each $|\alpha\_{p,j}|$ is bounded by $p^{1/2}$, hence if $\pi$ is an automorphic representation whose Rankin-Selberg powers $\pi\otimes\dots\otimes\pi$ are all automorphic, then the Satake parameters of $\pi$ at every unramified prime lie on the unit circle. This simple observation is one of the motivations for the Langlands program.


