# A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$

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.

Thank you.

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.
• 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 May 15, 2020 at 23:52
• 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!) May 16, 2020 at 2:19