  Motivated by string theory scattering amplitudes that are invariant under a
discrete U-duality, we study Fourier coefficients of Eisenstein series on
Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_9(R)$,
$E_{10}(R)$ and $E_{11}(R)$ corresponding to certain degenerate principal
series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show
that these Eisenstein series have very simple Fourier coefficients as expected
for their role as supersymmetric contributions to the higher derivative
couplings $R^4$ and $\partial^{4} R^4$ coming from 1/2-BPS and 1/4-BPS
instantons, respectively. This suggests that there exist minimal and
next-to-minimal unipotent automorphic representations of the associated
Kac-Moody groups to which these special Eisenstein series are attached. We also
provide complete explicit expressions for degenerate Whittaker vectors of
minimal Eisenstein series on $E_6(R)$, $E_7(R)$ and $E_8(R)$ that have not
appeared in the literature before.
