Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

• 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.

Subject headings

NATURVETENSKAP  -- Matematik -- Algebra och logik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Algebra and Logic (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

wave-front set

Automorphic function

Whittaker support

minimal representation

small representations

nilpotent orbit

next-to-minimal representation

string theory

Whittaker coefficient

Fourier coefficient

Publication and Content Type

art (subject category)

ref (subject category)