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A normalized cuspidal newform $f$ (either holomorphic or Maass) can be identified with a function on $\phi: \operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and it generates an irreducible cuspidal automorphic representation in some $L^2$-space. The central character of the corresponding representation depends on the some choice made in the transfer from $f$ to $\phi$.

Do these representations exhaust the cuspidal spectrum of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$?

Peter Humphries
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    Yes. This is probably buried somewhere in Gelbart, or Bump, or Goldfeld-Hundley (though I don't know where off the top of my head). – Peter Humphries Mar 21 '20 at 15:42
  • Note that this bijection only includes Maass forms of weight zero or one, and that weight one Maass forms with Laplacian eigenvalue $1/4$ are also holomorphic (their archimedean component is a limit of discrete series, namely the principal series $\mathrm{sgn} \boxplus 1$). – Peter Humphries Mar 21 '20 at 15:44
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    One can also define Maass forms of arbitrary integral weight (see e.g. Chapter 3 of Goldfeld-Hundley), and that one can map these to other Maass forms of different weights via raising and lowering operators. The point is that these are only newforms in the archimedean aspect if the weight is $0$, $1$, or an integer $\geq 2$ and additionally in the latter case that the form corresponds to a holomorphic cusp form. (You saw me give a talk about this in Pittsburgh.) – Peter Humphries Mar 21 '20 at 15:47
  • Hello @PeterHumphries , I think you are talking about Theorem 5.19 from the book by Gelbart (Automorphic forms on Adele group). This theorem says that there is a one to one correspondence between newforms and cuspidal automorphic representations of ${\mathrm{GL}}2(\mathbb{A}{\mathbb{Q}})$. I do not understand how Maass forms are coming into the picture. What am I missing here? – user15243 Nov 11 '23 at 17:58
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    @user15243 By newform, they mean either a holomorphic newform or a Maass newform. – Peter Humphries Nov 11 '23 at 20:35

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