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

TL;DR

This paper classifies reductive dual pairs within the algebraic group $PGL(n, ext{C})$, extending the understanding of dual pairs beyond classical matrix groups to a broader algebraic group context.

Contribution

It provides the first complete classification of reductive dual pairs in $PGL(n, ext{C})$, filling a gap in the theory of algebraic group dual pairs.

Findings

Complete classification of reductive dual pairs in $PGL(n, ext{C})$

Extension of dual pair theory beyond classical matrix groups

Clarification of the structure of dual pairs in algebraic groups

Abstract

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 $PGL(n,\mathbb{C})$.