EDIT July 22nd 2013: I add further details in bolded sentences:

Assuming Selberg's orthonormality conjecture and following Automorphisms of the Selberg class, I define automorphisms of the Selberg class $\mathcal{S}$ as bijective maps $f$ from $\mathcal{S}$ to itself such that:

1) $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,

2) $f$ maps a function of degree $d$ to a function of degree $d$,

3) for every $F$ in $\mathcal{S}$, $n_{f(F)}=n_F$, where $n_F $is the integer involved in Selberg's conjecture A,

4) if $F=F_{1}^{e_1}.F_{2}^{e_2}.F_{k}^{e_k}$, then $f(F)=f(F_1)^{e_1}.f(F_2)^{e_2}....f(F_k)^{e_k}$ (so that $f$ is "completely multiplicative").

The set of all such maps forms a group $Aut(\mathcal{S})$ under composition, the structure of which has been determined by Denis Chaperon de Lauzières in the link given above.

Now let $F$ be an element of $\mathcal{S}$ and let's define $G_{F}$ as the subgroup of $Aut(\mathcal{S})$ whose every element preserves $F$.
Is it true that $F$ is a primitive element of $\mathcal{S}$ of degree $d$ if and only if there exists a vector space $V$ over a finite extension of the $l$-adic field $\mathbb{Q}_{\ell}$ for some $l\in\mathbb{P}$ and a group morphism $\rho: G_{F}\to GL(V)$ such that $(V,\rho)$ is an irreducible representation of $G_{F}$ of degree $d$?

Moreover, considering that $G_{F}$ is somewhat analogous to the absolute Galois group of a field $\mathbb{K}$, the set $\{F^{n},n\ge 0\}$ playing the role of this field and $\mathcal{S}$ the role of its separable closure, and thus seeing $\rho$ as some kind of $\ell$-adic Galois representation, is it possible to associate to $\rho$ an automorphic representation $\pi_{\rho}$ such that for all $s$ $L(\rho,s)=L(\pi_{\rho},s)$? If so, does there exist an automorphism of $\mathcal{S}$ $\sigma_{\rho}$ such that $\sigma_{\rho}(F)(s)=L(\rho,s)$ for all $s$?

Thanks in advance.


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