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Theta liftings – a comparison between classical and representation theoretic results. (English) Zbl 0995.11033

Author’s introduction: Theta liftings have been considered both from a classical and from a representation theoretic point of view.
In the classical setting, one considers holomorphic theta series attached to integral quadratic forms as Siegel (or Hilbert-Siegel) modular forms. One is then interested in a description of the linear relations between the members of a given set of such theta series and in a characterization of the space of modular forms spanned by these theta series. The theta series in such a set are usually quite restricted in type, e.g., they belong to full lattices of some fixed level or they are theta series with characteristic (thetanullwerte) attached to a single lattice but with varying characteristic.
The representation theoretic approach considers the more general theta correspondence between automorphic forms on adelic orthogonal and symplectic (or metaplectic) groups defined using the oscillator (or Weil-) representation of the metaplectic group. Here one discusses for an irreducible representation space of automorphic forms on one of the groups whether it is in the image under the theta correspondence of a representation space of automorphic forms on the other group respectively whether its image under the correspondence is zero or not.
Although both types of questions appear to be extremely similar, they are not quite the same. It is the purpose of this note to discuss some cases in which a transfer of results between the two settings is made possible by recent results of C. Moeglin [J. Lie Theory 7, 231-238 (1997; Zbl 0885.22020)] and to describe some of the difficulties that occur in other cases.

MSC:

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

Citations:

Zbl 0885.22020
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