  We investigate the occurrence of divergences in maximal supergravity in
various dimensions from the point of view of supersymmetry constraints on the
U-duality invariant threshold functions defining the higher derivative
couplings in the effective action. Our method makes use of tensorial
differential equations that constrain these couplings. We study in detail the
Fourier modes and wave-front sets of the associated automorphic functions and
find that they are always associated to special nilpotent orbits.
