Fourier coefficients of minimal and nexttominimal automorphic representations of simplylaced groups
Abstract
In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simplylaced group. Let $\pi$ be a minimal or nexttominimal 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 PiatetskiShapiroShalika formula for cusp forms on $GL_n$. 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 nexttominimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.08296
 Bibcode:
 2019arXiv190808296G
 Keywords:

 Mathematics  Number Theory;
 High Energy Physics  Theory;
 Mathematics  Representation Theory
 EPrint:
 46 pages, this paper builds upon and extends the results of the second half of arXiv:1811.05966v1, which was split into two parts. The first part (with new title) is arXiv:1811.05966v2 and the present paper is an extension of the second part