Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
Artikel i vetenskaplig tidskrift, 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.

minimal representation

Whittaker support

Fourier coefficient

string theory

small representations

nilpotent orbit

Automorphic function

wave-front set

Whittaker coefficient

next-to-minimal representation

Författare

Dmitry Gourevitch

Weizmann Institute of Science

Henrik Gustafsson

Stanford University

Institute for Advanced Study

Rutgers University

Chalmers, Matematiska vetenskaper, Algebra och geometri

Axel Kleinschmidt

Max-Planck-Gesellschaft

Daniel Persson

Chalmers, Matematiska vetenskaper, Algebra och geometri

Siddhartha Sahi

Rutgers University

Canadian Journal of Mathematics

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

Vol. 74 1 122-169

Små automorfa representationer

Vetenskapsrådet (VR) (2018-04760), 2019-01-01 -- 2022-12-31.

Ämneskategorier (SSIF 2011)

Algebra och logik

Matematisk analys

DOI

10.4153/S0008414X20000711

Mer information

Senast uppdaterat

2025-03-17