On duality and negative dimensions in the theory of Lie groups and symmetric spaces

TL;DR

This paper explores duality and negative dimensions in Lie groups and symmetric spaces, interpreting symbolic formulas through Casimir operators and extending relations via Macdonald duality for symmetric functions.

Contribution

It provides a new interpretation of duality formulas in Lie groups and extends these relations to symmetric spaces using Macdonald duality.

Findings

Interpretation of $U(-N)=U(N)$ and $Sp(-2N)=SO(2N)$ via Casimir operators

Extension of duality relations to classical symmetric spaces

Application of Macdonald duality for symmetric functions

Abstract

We give one more interpretation of the symbolic formulae $U(-N)=U(N)$ and $Sp(-2N)=SO(2N)$ by comparing the values of certain Casimir operators in the corresponding tensor representations. We show also that such relations can be extended to the classical symmetric spaces using Macdonald duality for Jack and Jacobi symmetric functions.