Dual pairs in complex classical groups and Lie algebras
Marisa Gaetz
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 $PSO(n,\mathbb{C})$).