$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\mathfrak{g}}$ 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.

  • 1
    $\begingroup$ Since the various statements can be compartmentalized, a certain style of exposition might not combine them, since "it's obvious". Also, it is most popular to completely decompose 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
  • $\begingroup$ I think I agree with Paul---I'm not really sure how the $L^2$ decomposition and the $\mathcal A_0$ decomposition you write down are really that different, except possibly that you're asserting finite multiplicity of cuspidal representations in the latter space. Can you perhaps clarify? $\endgroup$ – Kimball Apr 6 '18 at 15:26

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