Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors

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 [1]. 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 derivative(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.