# Modular forms, Maass forms and Automorphic representations

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

• 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". – GH from MO Apr 4 '18 at 17:57
• @GHfromMO Many thanks for the references and enlightening comments! – Automorphic Apr 5 '18 at 12:53

• 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 $$\lambda$$)