Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

TL;DR

This paper classifies reductive dual pairs in orthosymplectic Lie supergroups and superalgebras, demonstrating their properties and invariance in Weyl-Clifford algebras, and establishing Howe duality for specific dual pairs.

Contribution

It provides a complete classification of reductive dual pairs in orthosymplectic Lie supergroups and superalgebras, and proves their invariance properties and Howe duality.

Findings

Full classification of reductive dual pairs in Lie superalgebras and supergroups.

Superalgebra of invariants generated by the dual Lie superalgebra.

Verification of Howe duality for specific orthosymplectic dual pairs.

Abstract

In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(E)$ on the Weyl-Clifford algebra $\textbf{WC}(E)$, we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((G\,, \mathfrak{g})\,, (G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(E)$, the superalgebra $\textbf{WC}(E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in $\textbf{WC}(E)$ is generated by the Lie superalgebra $\mathfrak{g}'$. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{spo}(\mathrm E)$ and the Lie supergroup $\textbf{SpO}(\mathrm E)$. Using this classification we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((\mathrm G\,, \mathfrak{g})\,, (\mathrm G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(\mathrm E)$, the superalgebra $\textbf{WC}(\mathrm E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in the Weyl-Clifford algebra $\textbf{WC}(\mathrm E)$, equipped with the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(\mathrm E)$, is generated by the Lie superalgebra $\mathfrak{g}'$. As an application, we prove that Howe duality holds for the dual pairs $({\textbf{SpO}}(2n|1)\,, {\textbf{OSp}}(2k|2l)) \subseteq {\textbf{SpO}}(\mathbb{C}^{2k|2l} \otimes \mathbb{C}^{2n|1})$.