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.
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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 non-cuspidal 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 Piatetski-Shapiro:
$\varphi$ is globally generic, that is, has non-vanishing Whittaker-Fourier 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 L-functions 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 Piatetski-Shapiro theorem:
- J. Shalika, The multiplicity one theorem for $\mathrm{GL}_n$ (1974)
- I. Piatetski-Shapiro, Multiplicity one theorems (1979)
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$\begingroup$ Perhaps giving some references (notes by Cogdell etc.) could help the OP (I am in a hurry, so I cannot compile a list now). $\endgroup$ Commented Oct 21, 2015 at 14:55
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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 Siegel-type 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..." $\endgroup$ Commented Oct 21, 2015 at 21:16
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$\begingroup$ 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. $\endgroup$ Commented Oct 22, 2015 at 7:34