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The local theta correspondence of irreducible type 2 dual reductive pairs. (English) Zbl 0842.11023

Let \(F\) be a non-archimedean local field. The author considers the dual reductive pair \((\text{GL}_n\), \(\text{GL}_{n+1})\) over \(F\). The purpose of this paper is to determine the image of the theta correspondence from \(\text{GL}_n\) to \(\text{GL}_{n+1}\). The author gives a result in this direction which partially describes the image in terms of the Bernstein-Zelevinsky classification of representations of the general linear groups. The proof involves a delicate local analysis of the Weil representation in this case and in particular makes essential use of the local theory of zeta functions.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11F27 Theta series; Weil representation; theta correspondences
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