Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition

Abstract

We determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg. This also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The key ingredient is a root-theoretic result. To prove it, we introduce the notion of the descending decomposition, which expresses every Weyl group element as a product of simple reflections in a certain way. It is suitable for induction and allows us to translate the question into a combinatorial statement.

References in corpus (4)

• Eisenstein series and automorphic representations
• Generalized and degenerate Whittaker models
• Fourier Coefficients for Theta Representations on Covers of General Linear Groups
• Raising nilpotent orbits in wave-front sets

Cited by in corpus (7)

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• Doubling Constructions: the complete L-function for coverings of the symplectic group
• On Fourier Coefficients of GL(n)-Automorphic Functions over Number Fields
• Doubling Constructions and Tensor Product $L$-Functions: the linear case
• Dimensions of Automorphic Representations, $L$-Functions and Liftings
• On the crystal graph description of the stable Weyl group multiple Dirichlet series