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I was reading adelization of classical automorphic forms and learnt that each cusp form corresponds to an automorphic representation of $\operatorname{GL}_n(\mathbb{A}_\mathbb{Q})$. I understood the proof. But then I found a statement that there is a one-to-one correspondence between newforms of the congruence subgroup $\Gamma_1(N)$ and the irreducible cuspidal representations of $\operatorname{GL}_n(\mathbb{A}_\mathbb{Q})$.

But I couldn't find any proof of it.
Please suggest some references.

Thank you.

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Newform theory for $\mathrm{GL}_n$ was originally developed over non-archimedean local fields (at least for $n\geq 3$). The local statements readily yield their global adelic counterparts, since a cuspidal automorphic representation is uniquely a restricted tensor product of local admissible generic representations. It is then straightforward to translate from adelic language to classical language.

So all you need is newform theory for admissible generic representations of $\mathrm{GL}_n$ over a non-archimedean local field. The guiding questions are: what is a newvector in this context, to what extent is it unique, how does it generate the representation, and so on. This theory was developed by Jacquet, Piatetski-Shapiro, and Shalika in their paper "Conducteur des représentations du groupe linéaire", Math. Ann. 256 (1981), 199-214. The paper also has a corrigendum.

A more accessible reference is Section 13.8 in the textbook of Goldfeld and Hundley "Automorphic representations and L-functions for the general linear group" (volume 1; volume 2). In general, reading this book should clarify many of the questions you might have. It filled an important gap in the literature.

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    $\begingroup$ Other useful references: this paper of Casselman for $\mathrm{GL}_2$: doi.org/10.1007/BF01428197. And another correct proof for $\mathrm{GL}_n$ due to Matringe: math.uni-bielefeld.de/documenta/vol-18/37.pdf $\endgroup$ May 15, 2020 at 23:52
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    $\begingroup$ @PeterHumphries: Thank you, Peter. Perhaps you can also mention somewhere the archimedean analogue you were developing. I am too busy now, e.g. I wrote the above answer in a rush! $\endgroup$
    – GH from MO
    May 16, 2020 at 2:02
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    $\begingroup$ Sadly still developing (at least for $\operatorname{GL}_n(\mathbb{R})$; the proofs are complete for $\operatorname{GL}_n(\mathbb{C})$). The theory itself takes too much work to state in a comment, so I will just link to the Oberwolfach Report where I first announced the result: mfo.de/document/1736/OWR_2017_40.pdf (P.S. It's 4am - go back to sleep!) $\endgroup$ May 16, 2020 at 2:19
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    $\begingroup$ All done now! See arxiv.org/abs/2008.12406 $\endgroup$ Aug 31, 2020 at 7:14

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