On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups

Dihua Jiang, Baiying Liu

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13 Scopus citations

Abstract

Fourier coefficients of automorphic representations π of Sp2n(A) are attached to unipotent adjoint orbits in Sp2n(F), where F is a number field and A is the ring of adeles of F. We prove that for a given π, all maximal unipotent orbits that give nonzero Fourier coefficients of π are special, and prove, under a well-acceptable assumption, that if π is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is F-anisotropic as algebraic group over F. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if F is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work.

Original languageEnglish (US)
Pages (from-to)343-389
Number of pages47
JournalJournal of Number Theory
Volume146
Issue numberC
DOIs
StatePublished - Jan 1 2015

Keywords

  • Automorphic forms
  • Fourier coefficients
  • Primary
  • Secondary
  • Unipotent orbits

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