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

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.