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Jacobi forms and some elements of the representation theory of the Jacobi group. (Formes de Jacobi et quelques éléments de la théorie des représentations du groupe de Jacobi.) (French) Zbl 0623.10015

Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 265-269 (1987).
[For the entire collection see Zbl 0605.00004.]
Two well known facts on automorphic forms on \(\mathrm{Sl}_2(\mathbb R)\) are the complete reducibility of \(L^2_0(\mathrm{Sl}_2(\mathbb Z)\setminus \mathrm{Sl}_2(\mathbb R))\) \((= \)cuspidal part of \(L^2(\mathrm{Sl}_2(\mathbb Z)\setminus \mathrm{Sl}_2(\mathbb R))\) under the regular representation and a “duality theorem”, which says that the dimension of \(S^k\) \((= \) space of holomorphic cusp forms of weight \(k\) on the upper half plane) is equal to the multiplicity of a discrete series representation in \(L^2_0(\mathrm{Sl}_2(\mathbb Z)\setminus \mathrm{Sl}_2(\mathbb R))\) – see e.g. the book of S. S. Gelbart [Automorphic forms on adèle groups (Ann. Math. Stud. 83) (1975; Zbl 0329.10018)].
The author describes (without proofs) that these results carry over to the theory of Jacobi modular forms, which was established by M. Eichler and D. Zagier [The theory of Jacobi forms (Prog. Math. 55) (1985; Zbl 0554.10018)]. This is remarkable, since the Jacobi group is an example of a non-reductive group.

MSC:

11F50 Jacobi forms
11F27 Theta series; Weil representation; theta correspondences
11F70 Representation-theoretic methods; automorphic representations over local and global fields