# Do we know absolute bounds for the norm of Satake parameters?

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 $$a=b=1$$.

• 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. – GH from MO Mar 8 at 18:22
• I'm interested in the Satake parameters whose norm equals $1$ assuming Ramanujan conjecture. – Sylvain JULIEN Mar 8 at 20:21
• We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. – GH from MO Mar 8 at 20:37
• Thank you for your comment. Feel free to turn it into an answer so that I can accept it and the question get closed. – Sylvain JULIEN Mar 8 at 20:48
• 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. – Subhajit Jana Mar 9 at 9:27

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.