Jacobi forms and discrete series representations of the Jacobi group. (English) Zbl 0695.10024

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.
Reviewer: Aloys Krieg


11F50 Jacobi forms
11F55 Other groups and their modular and automorphic forms (several variables)
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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