On the automorphic theta representation for simply laced groups
type: Item
instance of: scholarly article
title: On the automorphic theta representation for simply laced groups
zbMATH Open document ID: 0881.11050
DOI: 10.1007/BF02773635
author: David Ginzburg
author: Steve Rallis
author: David Soudry
published in: Israel Journal of Mathematics
publication date: 1997-11-06T00:00:00Z
review text: This paper is devoted to the investigation of certain automorphic representations of certain split, semisimple algebraic groups over a number field \(k\). The groups in question are to be of type \(D_m\) or \(E_m\). It was first shown by \textit{D. Kazhdan} and \textit{G. Savin} [Isr. Math. Conf. Proc. 2, 209-223 (1990; Zbl 0737.22008)] that there exists a ``very small'' representation of these groups and that this representation is a quotient of a principal series representation. The purpose of this paper is mainly to show that analogous representations exist globally in the cases listed and that they can be obtained as ``residues of Eisenstein series''. These remarkable representations are analogues of the classical theta representations. The authors consider the Fourier coefficients with respect to the unipotent radical of a certain parabolic subgroup (defined ad hoc) and show that these are often zero and have stronger invariance properties than one would expect. Furthermore the authors can show that there is a multiplicity one theorem in the case of \(D_m\). Finally, they give some indications as to how the emerging theory of these representations would lead to novel and intriguing analogues of the theta correspondences which arise between representations of elements of dual reductive pairs in symplectic groups, or their double covers.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1062357
zbMATH Keywords: simply laced groups
zbMATH Keywords: automorphic representations
zbMATH Keywords: semisimple algebraic groups
zbMATH Keywords: Eisenstein series
zbMATH Keywords: theta representations
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773635
OpenAlex ID: W2022162306
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description: scientific article; zbMATH DE number 1062357