Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
Journal article, 2022

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let ππ be a minimal or next-to-minimal automorphic representation of G. We prove that any η∈πη∈π 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 GLnGLn . 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 D5D5 and E8E8 with a view toward applications to scattering amplitudes in string theory.

wave-front set

Automorphic function

Whittaker support

minimal representation

small representations

nilpotent orbit

next-to-minimal representation

string theory

Whittaker coefficient

Fourier coefficient


Dmitry Gourevitch

Weizmann Institute of Science

Henrik Gustafsson

Rutgers University

Stanford University

Institute for Advanced Study

Chalmers, Mathematical Sciences, Algebra and geometry

Axel Kleinschmidt

Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Daniel Persson

Chalmers, Mathematical Sciences, Algebra and geometry

Siddhartha Sahi

Rutgers University

Canadian Journal of Mathematics

0008-414X (ISSN) 1496-4279 (eISSN)

Vol. 74 1 122-169

Små automorfa representationer

Swedish Research Council (VR) (2018-04760), 2019-01-01 -- 2022-12-31.

Subject Categories

Algebra and Logic

Mathematical Analysis



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