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Mathematics, Number Theory, math.NT,High Energy Physics - Theory, hep-th,Mathematics, Representation Theory, math.RT,
Abstract:
In this paper we analyze Fourier coefficients of automorphic forms on adelic
reductive groups $G(\mathbb{A})$. Let $\pi$ be an automorphic representation of
$G(\mathbb{A})$. It is well-known that Fourier coefficients of automorphic
forms can be organized into nilpotent orbits $\mathcal{O}$ of $G$. We prove
that any Fourier coefficient $\mathcal{F}_\mathcal{O}$ attached to $\pi$ is
linearly determined by so-called 'Levi-distinguished' coefficients associated
with orbits which are equal or larger than $\mathcal{O}$.
When $G$ is split and simply-laced, and $\pi$ is a minimal or next-to-minimal
automorphic representation of $G(\mathbb{A})$, we prove that any $\eta \in \pi$
is completely determined by its standard Whittaker coefficients with respect to
the unipotent radical of a fixed Borel subgroup, analogously to the
Piatetski-Shapiro--Shalika formula for cusp forms on $\mathrm{GL}_n$. In this
setting we also derive explicit formulas expressing any maximal parabolic
Fourier coefficient in terms of (possibly degenerate) standard Whittaker
coefficients for all simply-laced groups. We provide detailed examples for when
$G$ is of type $D_5$, $E_6$, $E_7$ or $E_8$ with potential applications to
scattering amplitudes in string theory.
