  This paper concerns some novel features of maximal parabolic Eisenstein
series at certain special values of their analytic parameter s. These series
arise as coefficients in the R4 and D4R4 interactions in the low energy
expansion of scattering amplitudes in maximally supersymmetric string theory
reduced to D=10-d dimensions on a torus T^d, d<8. For each d these amplitudes
are automorphic functions on the rank d+1 symmetry group E_d+1. Of particular
significance is the orbit content of the Fourier modes of these series when
expanded in three different parabolic subgroups, corresponding to certain
limits of string theory. This is of interest in the classification of a variety
of instantons that correspond to minimal or next-to-minimal BPS orbits. In the
limit of decompactification from D to D+1 dimensions many such instantons are
related to charged 1/2-BPS or 1/4-BPS black holes with euclidean world-lines
wrapped around the large dimension. In a different limit the instantons give
nonperturbative corrections to string perturbation theory, while in a third
limit they describe nonperturbative contributions in eleven-dimensional
supergravity. A proof is given that these three distinct Fourier expansions
have certain vanishing coefficients that are expected from string theory. In
particular, the Eisenstein series for these special values of s have markedly
fewer Fourier coefficients than typical ones. The corresponding mathematics
involves showing that the wavefront sets of the Eisenstein series are supported
on only certain coadjoint nilpotent orbits - just the minimal and trivial
orbits in the 1/2-BPS case, and just the next-to-minimal, minimal and trivial
orbits in the 1/4-BPS case. Thus as a byproduct we demonstrate that the
next-to-minimal representations occur automorphically for E6, E7, and E8, and
hence the first two nontrivial low energy coefficients are exotic
theta-functions.
