Symplectic supercuspidal representations and related problems
type: Item
instance of: scholarly article
title: Symplectic supercuspidal representations and related problems
zbMATH Open document ID: 1253.11061
author: Chufeng Nien
author: Yujun Qin
author: Dihua Jiang
published in: Science China. Mathematics
publication date: 2010-05-26T00:00:00Z
review text: In this work, the authors summarize results (originally obtained by them, and many other people) related to symplectic supercuspidal representations of \(\text{GL}(2n)\) over a non-archimedean field of characteristic zero. In more detail, they discuss the existence of certain models (Shalika models, linear models, Gelfand-Graev models) on the representations of the general linear groups and related classical and metaplectic groups (related to general linear groups via parabolic induction or arising as a stabilizers of certain functionals on unipotent radicals). The existence of these models is closely related with, in part proved, in part conjectural Langlands functoriality transfer between general linear and classical (or metaplectic) groups and theta correspondences. The symplecticity condition (given initially in terms of the existence of the pole of the exterior square \(L\)-function for \(s=0\)) ensures that these cuspidal representations of general linear groups are in the image of the appropriate functorialities. We now briefly describe the content of the paper. In the first section some historical and introductory material is presented. In the second section the Shalika model (for an irreducible admissible representation of \(\text{GL}(2n)\)) is introduced (as a possible embedding in a representation induced from the character of the Shalika subgroup), and the local uniqueness of this model is stated. Then the generalized Shalika model for \(\text{SO}(4n)\) is introduced and these two are related by the parabolic induction, at least for supercuspidal representations and induction in the Siegel case, where the Levi subgroup is isomorphic to \(\text{GL}(2n)\) (as explained in Theorem 2.3; reducibility outside of zero in this induction is required). Then, another model for \(\text{GL}(2n)\) is introduced-namely the linear model, where an irreducible admissible representation of \(\text{GL}(2n)\) has a linear model if it can be embedded in the representation ind…
reviewed by: Marcela Hanzer
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 5712117
zbMATH Keywords: symplectic supercuspidal representations
zbMATH Keywords: Shalika models
zbMATH Keywords: linear models
zbMATH Keywords: Langlands functoriality
zbMATH Keywords: descent
MaRDI profile type: Publication
full work available at URL: s11425-010-0056-0
OpenAlex ID: W2008832569
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DOI: 10.1007/S11425-010-0056-0
description: scientific article; zbMATH DE number 5712117