Jacobi forms and discrete series representations of the Jacobi group
type: Item
instance of: scholarly article
title: Jacobi forms and discrete series representations of the Jacobi group
zbMATH Open document ID: 0695.10024
DOI: 10.1007/BF02570858
author: Rolf Berndt
author: Siegfried Böcherer
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183795
review text: The paper under review is a bridge between the classical way of dealing with Jacobi forms and the general representation-theoretical way of looking at automorphic forms. The authors study the Jacobi group \(G^ J=\mathrm{SL}_ 2(\mathbb R)\ltimes H(\mathbb R)\), which is an example of a non- reductive group. By virtue of the non-commutative structure of the algebra of invariant differential operators this case is more complicated than the theory for reductive groups. Nevertheless the authors can derive results similar to the general representation theory of \(\mathrm{SL}_ 2\). Considering \(\Gamma^ J=\mathrm{SL}_ 2(\mathbb Z)^ 2\ltimes \mathbb Z^ 2\) Jacobi forms can be defined as functions on \(G^ J\). The cusp condition can be described by introducing a convenient generalization of cuspidal subgroups from \(\mathrm{SL}_ 2\) to \(G^ J\). Here the cusp \(i\infty\) of the modular group \(\mathrm{SL}_ 2(\mathbb Z)\) degenerates under the influence of the Heisenberg group. The subspace \({\mathcal H}=L^ 2_ 0(\Gamma^ J\setminus G^ J)\) of \(L^ 2(\Gamma^ J\setminus G^ J)\) is given by the cusp condition. The right regular representation on \({\mathcal H}\) has a discrete spectrum. Finally a duality theorem is derived, which connects the multiplicities of discrete series representations with dimensions of spaces of Jacobi cusp forms.
reviewed by: Aloys Krieg
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 4137872
zbMATH Keywords: Jacobi forms
zbMATH Keywords: Jacobi group
zbMATH Keywords: non-reductive group
zbMATH Keywords: invariant differential operators
zbMATH Keywords: duality theorem
zbMATH Keywords: multiplicities of discrete series representations
zbMATH Keywords: dimensions of spaces of Jacobi cusp forms
MaRDI profile type: Publication
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OpenAlex ID: W1992476193
description: scientific article; zbMATH DE number 4137872