Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

TL;DR

This paper derives formulas for denominator identities of basic classical Lie superalgebras, links positive root sets to dual pairs in Lie groups, and recovers the Theta correspondence for compact dual pairs.

Contribution

It introduces explicit denominator formulas for Lie superalgebras and connects root systems to dual pairs, advancing the understanding of their representation theory.

Findings

Formulas for denominator and superdenominator of Lie superalgebras

Connection between positive roots and reductive dual pairs

Recovery of the Theta correspondence for compact dual pairs

Abstract

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.