  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 [arXiv:1111.2983].
