Abstract
AbstractIn 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.
Highlights
IntroductionE simplest case of a nonabelian U is one that admits a Heisenberg structure, i.e., [U , U] is a onedimensional group, and this will be an important tool for us when we analyze groups of type E that do not admit any abelian unipotents U as radicals of parabolic subgroups
Introduction and main results1.1 IntroductionLet K be a number field and A = AK = ∏′ Kν its ring of adeles
It is interesting to ask which Fourier coefficients are Eulerian [Gin, Gin ]. e expectation, based on the reduction formula of [FKP ] for Eisenstein series and explicit examples checked there, is that Whittaker coefficients Wφ[η] of an Eisenstein series η on a group G are Eulerian if the orbit of φ is lies in WS(η)
Summary
E simplest case of a nonabelian U is one that admits a Heisenberg structure, i.e., [U , U] is a onedimensional group, and this will be an important tool for us when we analyze groups of type E that do not admit any abelian unipotents U as radicals of parabolic subgroups In this case, the lower central series coincide with the derived series. We will analyze Fourier coefficients and expansions in the case of special classes of automorphic forms on split, -laced Lie groups. We apply the techniques of [GGK+] to relate maximal parabolic Fourier coefficients, which are hard to compute, to a more manageable class of coefficients such as the known Whittaker coefficients with respect to the unipotent radical of a Borel subgroup. We discuss the class of Fourier coefficients studied in [GGS , GGS, GGK+]. is class includes parabolic coefficients, coefficients of lower central series (but not the derived series) for unipotent radicals of parabolics, and the coefficients considered in [GRS , Gin , Gin , JLS ]
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