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I am beginning to learn about automorphic forms, and stay perplex concerning the two languages of "forms" versus "representations" often used at the same time. As far as I understand,

  • a modular/Maass form generates an automorphic representation (by considering the space of its right translations)
  • conversely, an automorphic representation gives a unique automorphic form (as a newvector, i.e. a nontrivial element in $\pi^K$ which is one dimensional for a suitable (?) compact subgroup $K$)

I would like to understand, where does the distinction between modular form and Maass form appear in the representation language?

Thanks in advance for any reference or explanation

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    $\begingroup$ Some comments: 1. You are talking here about automorphic representations of $\mathrm{GL}_2$ over $\mathbb{Q}$. Other groups and other number field come with additional complications. 2. You need to start with a Hecke newform (holomorphic or Maass) to get an (irreducible cuspidal) automorphic representation (by taking the Hilbert space closure of its right translations). 3. The answer of Desiderius Severus is perfectly fine. For this material, I recommend Bump's book "Automorphic forms and representations". $\endgroup$
    – GH from MO
    Commented Apr 4, 2018 at 17:57
  • $\begingroup$ @GHfromMO Many thanks for the references and enlightening comments! $\endgroup$
    – Automorphic
    Commented Apr 5, 2018 at 12:53

1 Answer 1

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The archimedean component of the representation determines the type of the underlying automorphic form. Loosely speaking,

  • if $\pi_\infty$ is a discrete series (of index $k$), the underlying form is an holomorphic cusp form (of weight $k$, and level the arithmetic conductor of $\pi_f$)
  • if $\pi_\infty$ is a principal or complementary series (of index $\lambda$), the underlying form is a Hecke-Maass cusp form (of eigenvalue $\lambda$)

The correspondence is detailed and proved in Gelbart, Automorphic forms on Adele Groups, 5.C "Some explicit features of the correspondence between cusp forms and representations".

Maybe you should look at the more form-casted answer given here, and the associated reference in Bump.

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