  We investigate Fourier coefficients of automorphic forms on split
simply-laced Lie groups G. We show that for automorphic representations of
small Gelfand-Kirillov dimension the Fourier coefficients are completely
determined by certain degenerate Whittaker vectors on G. Although we expect our
results to hold for arbitrary simply-laced groups, we give complete proofs only
for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates
Fourier coefficients of automorphic forms with nilpotent orbits of G. Our
results complement and extend recent results of Miller and Sahi. We also use
our formalism to calculate various local (real and p-adic) spherical vectors of
minimal representations of the exceptional groups E_6, E_7, E_8 using global
(adelic) degenerate Whittaker vectors, correctly reproducing existing results
for such spherical vectors obtained by very different methods.
