# A possible application of representation theory to Galois classes of L-functions

I define the notion of a Galois class of L-function as follows:

$A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true:

1) $A$ is a subset of the Selberg class containing the constant function equal to $1$
2) whenever $F$ and $G$ are elements of $A$, then so is $F.G$
3) every element of $A$ factors in a unique fashion as a product of primitive elements of $A$

Then I denote $M$ the maximal Galois class of L-functions (conjecturally equal to the whole Selberg class) and define its group of automorphisms (under composition) $Aut(M)$ as follows:
$\Phi$ is an automorphism of $M$ if and only if the following conditions simultaneously hold true:

1) $\Phi$ is a bijective map from $M$ to itself
2) $\forall F\in M$, $deg(\Phi(F))=deg(F)$, where $deg(F)$ is the degree of $F$ as an element of the Selberg class
3) $\forall (F,G)\in M^{2}$, $\Phi(F.G)=\Phi(F).\Phi(G)$
4) $\Phi$ maps a primitive element of $M$ to a primitive element of $M$

Given an element $F$ of $M$ of degree $d$, one can write $F=\prod_{i=1}^{n}F_{i}^{e_i}$ where $\forall 1\leqslant i\leqslant n$ $F_{i}$ is a primitive element of $M$ and $e_{i}$ is a positive integer. Let's consider the group $G_{F}$ (under composition) of automorphisms of $M$ preserving $F$.

My questions are:
Q1) Does there exist a $d$-dimensional semi-simple complex representation $(V,\rho)$ of $G_{F}$ such that $V$ is isomorphic to the direct sum $\bigoplus_{i=1}^{n}e_{i}V_{i}$ where $\forall 1\leqslant i\leqslant n$ $V_{i}$ is an irreducible complex representation of $G_{F}$ of degree $deg F_{i}$?
Q2) Given that, at least for finite groups, irreducible characters form an orthonormal basis of the vector space of central functions, is it possible to establish the equality $\langle\chi_{V_{i}},\chi_{V_{j}}\rangle=\lim_{x\to\infty}\dfrac{1}{\log\log x}\sum_{p\leqslant x,p\in\mathbb{P}}\dfrac{a_{p}(F_{i}).\overline{a_{p}(F_{j})}}{p}$, where $\langle\chi_{V_{i}},\chi_{V_{j}}\rangle=\dfrac{1}{\vert G\vert}\sum_{g\in G}\overline{\chi_{V_{i}}(g)}\chi_{V_{j}}(g)$ and $a_{p}(H)$ is the $p$-th coefficient of the Dirichlet series $\sum_{n>0}\dfrac{a_{n}}{n^{s}}$ defining $H(s)$ for $\Re(s)>1$?
Q3) Suppose $F$ is primitive. Are all elements of $G_{F}$ conjugates?

EDIT August 31rd 2013: I just had an idea of what the desired representation could be. As $F=G$ if and only if $\Lambda_{F}=\Lambda_{G}$ where $\Lambda_{F}$ is the complete L-function associated to $F$, to an element $g$ of $G_{F}$ one can associate an element $\sigma_{g}$ of $\mathfrak{S}_{m}$ where $m$ is the integer such that the gamma factor of $F$ is $\displaystyle{\gamma_{F}(s)=\vert w_{F}\vert Q_{F}^{s}\prod_{j=1}^{m}\Gamma(\omega_{j}^{F}s+\mu_{j}^{F})}$ (conjecturally $m=d$), with $\sigma_{g}$ verifying the following conditions:
C1) $\displaystyle{\sigma_{g}(\prod_{j=1}^{m}\Gamma(\omega_{j}^{F}s+\mu_{j}^{F}))}$ $=\displaystyle{\prod_{j=1}^{m}\sigma_{g}(\Gamma(\omega_{j}^{F}s+\mu_{j}^{F}))}$ $=\displaystyle{\prod_{j=1}^{m}\Gamma(\omega_{j}^{F}s+\mu_{j}^{F})}$

C2) $\sigma:G_{F}\to\mathfrak{S}_{m},\ \ g\mapsto \sigma_{g}$ is a group homomorphism.

Then one associates to $\sigma_{g}$ its permutation matrix $P_{\sigma_{g}}$, which is an element of $GL_{m}(\mathbb{C})$. Hence one can expect the equality $\rho(g)=P_{\sigma_{g}}$ to hold true.
I guess the next step consists in finding a way to prove that $m=d$...

EDIT September 10th 2013: the group homorphism $\sigma$ may be an isomorphism. Would a proof of this be useful to show that $m=d$?

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It seems one could provide at least a partial answer to Q1 by proving $G_{F}$ is profinite. So, is $G_{F}$ profinite? –  Sylvain JULIEN Aug 6 '13 at 12:26
I think the countability conjecture as stated in mat.unimi.it/users/molteni/research/papers-pdf/… implies $G_{F}$ is profinite. Let's write $\{P_{n}\}_{n>0}$ be the elements of the set of equivalence classes considered by the authors such that $\forall i$ no element of $P_{i}$ is a factor of $F$, and let's consider the inverse limit of groups $G_{n}(F)$ defined as follows: $G_{0}(F)$ is the trivial group, and for $n\geqslant 0$, $G_{n+1}(F)$ is the group of automrphisms of $A_{n}$ preserving $F$...(to be continued) –  Sylvain JULIEN Aug 6 '13 at 13:24
...where $A_{n+1}$ is the Galois class of L functions generated by $A_{n}\cup \{P_{n}\}$. Then the inverse limit of all the $G_{n}(F)$ should be equal to $G_{F}$, hence making this last group a profinite group. –  Sylvain JULIEN Aug 6 '13 at 13:30