Dual pairs in complex classical groups and Lie algebras

TL;DR

This paper classifies dual pairs of reductive Lie subalgebras and subgroups within complex classical groups and their Lie algebras, extending Howe's notion and making progress on the projective cases.

Contribution

It provides a comprehensive classification of dual pairs in complex classical groups and Lie algebras, and advances understanding of dual pairs in projective classical groups.

Findings

Classified dual pairs in complex classical groups and Lie algebras.

Made substantial progress in classifying dual pairs in projective classical groups.

Extended Howe's dual pair concept to new algebraic group contexts.

Abstract

In Roger Howe's 1989 paper, ``Remarks on classical invariant theory," Howe introduces the notion of a dual pair of Lie subalgebras: a pair $(\mathfrak{g}_1, \mathfrak{g}_2)$ of reductive Lie subalgebras of a Lie algebra $\mathfrak{g}$ such that $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are each other's centralizers in $\mathfrak{g}$. This notion has a natural analog for algebraic groups: a dual pair of subgroups is a pair $(G_1, G_2)$ of reductive subgroups of an algebraic group $G$ such that $G_1$ and $G_2$ are each other's centralizers in $G$. In this paper, we classify the dual pairs in the complex classical groups ($GL(n,\mathbb{C})$, $SL(n,\mathbb{C})$, $Sp(2n,\mathbb{C})$, $O(n,\mathbb{C})$, and $SO(n,\mathbb{C})$) and in the corresponding Lie algebras ($\mathfrak{gl}(n,\mathbb{C})$, $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{sp}(2n,\mathbb{C})$, and $\mathfrak{so}(n,\mathbb{C})$). We also present substantial progress towards classifying the dual pairs in the projective counterparts of the complex classical groups ($PGL(n,\mathbb{C})$, $PSp(2n,\mathbb{C})$, $PO(n,\mathbb{C})$, and $PSO(n,\mathbb{C})$).