$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in particular in connection with the Eichler-Shimura isomorphism. However I still couldn't find a source that treats all this concepts in the full generality of a number field to get precise definitions. I also feel I might be missing some key-words or misnaming some objects, for the sources I do find vary a lot in their description of what cusp forms are, though I'm sure the definitions therein are all equivalent in some sense. I'm looking for a (ideally single) reference that covers the statements numbered (1)-(5) below, or at least confirms that they are true.
Let $G$ be a reductive linear algebraic group and $\mathbb{A}$ the adèle ring of $F$. Following Piatetski-Shapiro's definition in the first Corvalis volume, a cusp form for $G(F)$ is a smooth function $f\colon G(F)\backslash G(\mathbb{A}) \to \mathbb{C}$ on which the center $C = Z(G(\mathbb{A}))$ acts through a character $\omega\colon C \to \mathbb{C}^\times$, the coefficients $|f(g)|$ are square integrable over $CG(F)\backslash G(\mathbb{A})$ and have vanishing integral over $U(F)\backslash U(\mathbb{A})$ where $U$ denotes the unipotent radical of any parabolic subgroup of $G$.
The smoothness condition means that $f$ can be thought of as a function $\Gamma\backslash G(\mathbb{A}_\infty) = \Gamma\backslash G(F\otimes_{\mathbb{Q}} \mathbb{R}) \to \mathbb{C}$ where $\Gamma$ is some congruence subgroup of $G(\mathcal{O}_F)$, and is from this connection that we recover the classical theory of modular forms from the automorphic case for $G = \PGL_2$. Thus, given any central character $\omega$ and congruence subgroup of $G(\mathcal{O}_F)$ or, which amounts to the same, a compact open subgroup $K_f$ of the finite àdelic part $G(\mathbb{A}^\infty)$, one defines the subspace $S(K_f,\omega)$ of the space of all cusps forms to be the set of all cusp forms constant on the cosets of $G(F)\backslash G(\mathbb{A}) /K_f$. In comparison with the modular case, $K_f$ plays the role of the "level" of the modular function.
Now assume $G = \PGL_n$ for some $n$, so that $\omega$ is trivial. As an "admissible representation" of $G(\mathbb{A})$, which really means a compatible representation of $G(K_\nu)$ for all finite places $\nu$ and a $(\mathfrak{g}_\nu, K_\nu)$-module for all infinite places $\nu$, we get a decomposition of $S(K_f)$ as a direct sum of closed irreducible subspaces, each isotypical component having multiplicity exactly one and being a bona fide real Lie representation $\pi$ of $G(A_\infty)$. The $\pi$'s that apear in this decomposition are called the irreducible automorphic cuspidal representations of $G(A_\infty)$.
If I understand correctly, this representations are usually not finite-dimensional (1).
However, apparently there is a unique finite-dimensional Lie representation $\rho = \rho(\pi)$ of $G(A_\infty)$ that induce the same infinitesimal character on center of the universal enveloping algebra $U(\mathfrak{g}_\infty) = \bigotimes_{\nu\text{ infinite}} U(\mathfrak{g}_\nu)$ (2).
And for $G = \PGL_2$, just so happens that this representation $\rho$ is exactly the tensor product of $\operatorname{Sym}^{\lambda_\nu} \mathbb{R}^2$ for $\nu$ a real place and $\operatorname{Sym}^{\lambda_\nu} \mathbb{C}^2 \otimes_{\mathbb{C}} \overline{\operatorname{Sym}^{\lambda_\nu} \mathbb{C}^2}$ for $\nu$ a complex place (3).
Hence, in the $G = \PGL_2$ case, the family of parameters $\lambda = (\lambda_\nu)$ describe completely the decomposition of $S(K_f)$ into irreducibles, and if $f \in S(K_f)$ lies in the isotypical factor $\pi$ whose associated $\rho$ has parameters $\lambda$, we say $f$ has weight $\lambda$ (4).
Let's denote the space of cusps forms of weight $\lambda$ by $S_\lambda(K_f)$. If $\Gamma$ is the congruence subgroup defined by $K_f$ and $F$ has $r_1$ real embeddings and $r_2$ conjugate pairs of complex embeddings, a consequence of the Eichler-Shimura is an equality of dimensions $$\dim_{\mathbb{C}} H^{r_1+r_2}_{\text{cusp}}(\Gamma, \rho) = 2^{r_1}\dim_{\mathbb{C}} S_{\lambda}(K_f)\,,$$ where the cuspidal cohomology on the left is intersection of the kernels of the restriction maps $\Gamma \to \Gamma\cap U(\mathbb{A}_\infty)$, where $U$ is again the unipotent radical of the parabolic subgroups of $G$ (5).
