Dual pairs in $PGL(n,\mathbb{C})$

In Roger Howe’s seminal 1989 paper “Remarks on classical invariant theory,” he introduces the notion of Lie algebra dual pairs, and its natural analog in the groups context: a pair $(G_1,G_2)$ of reductive subgroups of an algebraic group $G$ is a dual pair in $G$ if $G_1$ and $G_2$ equal each other’s centralizers in $G$. While reductive dual pairs in the complex reductive Lie algebras have been classified, much less is known about algebraic group dual pairs, which were only fully classified in the context of certain classical matrix groups. In this paper, we classify the reductive dual pairs in $PGL(n,\mathbb{C})$.