Fourier coefficients of automorphic forms, character variety orbits, and small representations

Text. We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G(2), we perform an analysis of the expansions on split real forms of E-6 and E-7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fourier coefficients on GL(n). We use this mechanism, for example, to show that the minimal representation of either E-6 or E-7 never occurs in the cuspidal automorphic spectrum. We also give a complete description of the internal Chevalley modules of all complex Chevalley groups - that is, the orbit decomposition of the Levi factor of a maximal parabolic on its unipotent radical. This generalizes classical results on trivectors and in particular includes a full description of the complex character variety orbits for all maximal parabolics. The results of this paper have been applied in the string theory literature to the study of BPS instanton contributions to graviton scattering (Green et al., 2011, [12]). Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ELkyOT8c28I. (C) 2012 Published by Elsevier Inc.