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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 $a=b=1$.

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  • $\begingroup$ The archimedean Satake parameters of a classical Maass form of Laplace eigenvalue $1/4+t^2$ are $\pm it$. The eigenvalues are unbounded, so the Satake parameters are also unbounded. So it is not clear what you are after. $\endgroup$
    – GH from MO
    Commented Mar 8, 2020 at 18:22
  • $\begingroup$ I'm interested in the Satake parameters whose norm equals $1$ assuming Ramanujan conjecture. $\endgroup$
    – Sylvain JULIEN
    Commented Mar 8, 2020 at 20:21
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    $\begingroup$ We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. $\endgroup$
    – GH from MO
    Commented Mar 8, 2020 at 20:37
  • $\begingroup$ Thank you for your comment. Feel free to turn it into an answer so that I can accept it and the question get closed. $\endgroup$
    – Sylvain JULIEN
    Commented Mar 8, 2020 at 20:48
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    $\begingroup$ If $\alpha_{p,j}:=p^{\mu_{p,j}}$ are bounded independent of $p$ then $\Re(\mu_{p,j})\le 0$ for all $j$. At least for $PGL(n)$ this implies that $\Re(\mu_{p,j})=0$ for all $j,p$, i.e. Ramanujan. $\endgroup$
    – Subhajit Jana
    Commented Mar 9, 2020 at 9:27

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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.

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