TY - JOUR
T1 - On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups
AU - Jiang, Dihua
AU - Liu, Baiying
N1 - Publisher Copyright:
© 2014 Elsevier Inc.
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Automorphic forms
KW - Fourier coefficients
KW - Primary
KW - Secondary
KW - Unipotent orbits
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U2 - 10.1016/j.jnt.2014.03.002
DO - 10.1016/j.jnt.2014.03.002
M3 - Article
AN - SCOPUS:84897352747
SN - 0022-314X
VL - 146
SP - 343
EP - 389
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - C
ER -