Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

MINIMAL AUTOMORPHIC POSETS AND THE PROJECTION PROPERTY

We present some results concerning the projection property for finite ordered sets. We show that sums of non-trivial ramified ordered sets over a connected poset of at least two elements are projective. We construct a family of minimal automorphic posets of reach 2 and length 2 and show they are projective.