Full theta lifting for type II reductive dual pairs

In this article, we study the full theta lifting for two cases of type II reductive dual pairs over a nonarchimedean local field. Firstly, we determine the structure of the full theta lifts of all irreducible representations for dual pair \((\mathrm{GL}(2),\mathrm{GL}(2))\). Secondly, we prove that the full theta lift of an irreducible tempered representation is irreducible and tempered for dual pair \((\mathrm{GL}(n),\mathrm{GL}(n+1))\), where \(n\) is a positive integer.