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MR2214128 (2007b:11068) 11F70 (22E55)
Ginzburg, David (IL-TLAV)
Certain conjectures relating unipotent orbits to automorphic representations. (English
Israel J. Math. 151 (2006), 323355.
One important problem in the theory of automorphic forms is to understand the Fourier coefficients
of automorphic representations. A central role in the theory is played by a certain set of unipotent
orbits of G(C), denoted by OG(), where G is a split classical group and is an automorphic
representation of G(A). To each unipotent orbit, one attaches a set of Fourier coefficients of . A
unipotent orbit O is in OG() if has a nonzero Fourier coefficient associated to O and if has
no nonzero Fourier coefficients associated to unipotent orbits greater than O (with respect to the
standard partial ordering). Knowledge of the structure of OG() is crucial for various applications.
The paper under review formulates many conjectures about the structure of OG(). Most of these
conjectures are illustrated with specific examples.
In the main section of the paper, OG() is studied when is an Eisenstein series or any of its
residues. Let M be a Levi part of a parabolic subgroup of G and an automorphic representation of