Annihilator of $({\mathfrak g},K)$-modules of ${\mathrm O}(p,q)$

TL;DR

This paper investigates the annihilators of specific $(\mathfrak{g},K)$-modules for the orthogonal group, showing they are the Joseph ideal only in the trivial case, with key roles played by Casimir elements.

Contribution

It demonstrates that the annihilator of certain $(\mathfrak{g},K)$-modules equals the Joseph ideal only when the module parameter is zero, revealing a new characterization.

Findings

Annihilator equals Joseph ideal only for m=0

Casimir elements are crucial in the proof

Provides a new criterion for module annihilators

Abstract

Let ${\mathfrak g}$ denote the complexified Lie algebra of $G={\mathrm O}(p,q)$ and $K$ a maximal compact subgroup of $G$. In the previous paper, we constructed $({\mathfrak g},K)$-modules associated to the finite-dimensional representation of ${\mathfrak sl}_2$ of dimension $m+1$, which we denote by $M^{+}(m)$ and $M^{-}(m)$. The aim of this paper is to show that the annihilator of $M^{\pm}(m)$ is the Joseph ideal if and only if $m=0$. We shall see that an element of the symmetric of square $S^{2}({\mathfrak g})$ that is given in terms of the Casimir elements of ${\mathfrak g}$ and the complexified Lie algebra of $K$ plays a critical role in the proof of the main result.