$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\mathfrak{g}}$ Let $\G$ be a semisimple group over $\Q$, and let $\A$ denote the adèle ring of $\Q$. Let $L^2_0(\G(\Q)\backslash\G(\A))$ denote the space of cuspidal $L^2$ functions and $\Autom_0(\G)$ denote the space of cuspidal automorphic forms for $\G$. Let $K$ be a maximal compact subgroup of $\G(\R)$ and $\cen$ be the centre of the universal enveloping algebra of the complexification of $\lieg = {\rm Lie}(\G(\R))$. (I did not take the group reductive with a central character to simplify notation, but feel free to comment about the more general case too)

In textbooks about automorphic forms, you find the following theorem of Gelfand and Piatetski-Shapiro : $$ L^2_0(\G(\Q)\backslash\G(\A)) \cong \widehat{\bigoplus_{\pi}}\pi^{m(\pi)} $$ as unitary representations of $\G(\A)$, where the RHS is a Hilbert direct sum over irreducible unitary representations of $\G(\A)$.

Unless I made a mistake, it seems to be an easy consequence of this theorem, the tensor product theorem and the definition of cuspidal automorphic forms that $$ \Autom_0(\G) \cong \bigoplus_{\pi = \pi_\infty\otimes\pi_f} (\pi_\infty^{\rm fin} \otimes \pi_f)^{m(\pi)} $$ as representations of $\G(\A)$, where the RHS is an algebraic direct sum, $\pi_f$ is an irreducible admissible representation of $\G(\A_f)$, $\pi_\infty$ is an irreducible unitary representation of $\G(\R)$ and $\pi_\infty^{\rm fin}$ denotes the $K$-finite, $\cen$-finite vectors in $\pi_\infty$.

Question 1: Is this correct?

It seems to me that this, if correct, is a nice way to explain the relevance of the decomposition of the $L^2$ space, but I have not found it in any textbook (which may mean that I missed a subtlety, hence the question). The only place I saw a similar statement was the last paragraph of Section 2 in the article Automorphic forms and automorphic representations by Borel and Jacquet in the 1979 Corvallis proceedings, but they did not state the adelic version.

Question 2: (assuming the answer to 1 is Yes) Is there a reference stating this result? If not, then why does nobody tell us about this decomposition?

I could see some potential reasons: it is not interesting, it is not the way automorphic forms are used, etc.

completelydecompose an irreducible unitary of the adele group, rather than just into archimedean and non-archimedean parts. But maybe this kind of thing is not what you are really asking about... $\endgroup$ – paul garrett Apr 5 '18 at 22:52