The local theta correspondence of irreducible type 2 dual reductive pairs (Q1908314)

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.