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

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 ${\mathfrak{s}l}_2$ 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.

13 pages