we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and Serre. My question is the following: have automorphic forms Fourier expansions? Thanks in advance.
1 Answer
I don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless.
Fourier expansion is a very important tool in the study of automorphic forms, and works for any automorphic form over any reductive algebraic group.
As usual, the cuspidal and noncuspidal cases are very different, and the $G=\mathrm{GL}_n(\mathbb{A})$ case is particularly well understood.
For instance, for $\mathrm{GL}_n(\mathbb{A})$ and $\varphi$ cuspidal, we have a beautiful theorem of Shalika and PiatetskiShapiro:
$\varphi$ is globally generic, that is, has nonvanishing WhittakerFourier coefficients, and the Fourier expansion is absolutely convergent and uniformly converges on any compact set.
For a nice exposition of the analytic theory on $\mathrm{GL}_n$, see
 J. W. Cogdell, Notes on Lfunctions for $\mathrm{GL}_n$
As a textbook reference, section 3 of Bump's classic book covers this aspect of automorphic forms:
 Daniel Bump, Automorphic Forms and Representations
A more concrete and advanced reference that I've found very useful:
The original references for the Shalika and PiatetskiShapiro theorem:
 J. Shalika, The multiplicity one theorem for $\mathrm{GL}_n$ (1974)
 I. PiatetskiShapiro, Multiplicity one theorems (1979)

$\begingroup$ Perhaps giving some references (notes by Cogdell etc.) could help the OP (I am in a hurry, so I cannot compile a list now). $\endgroup$ Oct 21, 2015 at 14:55

1

$\begingroup$ One should be careful: in groups other than split $GL_n$, not all automorphic forms/representation "have Whittaker models", e.g., holomorphic Siegel modular forms do not. Sure, there is still a "Fourier expansion" along the Siegeltype maximal proper parabolic, but there are many more Fourier coefficients than Hecke eigenvalues... So I'd say that the "yes" has to be "up to a point..." $\endgroup$ Oct 21, 2015 at 21:16

$\begingroup$ Thank you for your remark @paul garrett. I am actually interested in $\lambda_{Sym^rf}$ which is an automorphic form in $GL_{r+1}$ so the answer is certainly yes in this case. $\endgroup$ Oct 22, 2015 at 7:34