Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $V$ by $V_\sigma$, so that $V=\bigoplus_{\sigma\in\widehat{K}}V_\sigma$.
Question: Suppose that $V$ is generated by a $K$-invariant subspace $W\subseteq V_\sigma$ for some $\sigma$, i.e., $V=U(\mathfrak g)W$. Is it true that $V_\sigma=U(\mathfrak g)^KW$? Here $U(\mathfrak g)^K$ is the $K$-invariant part of the enveloping algebra $U(\mathfrak g)$ under the adjoint action of $K$.
This statement is true when $K$ is connected, and basically follows from Proposition 9.1.10 in Dixmier's book Enveloping Algebras. However, I am interested in the cases where $K$ is disconnected. In the latter cases, I cannot generalize Dixmier's argument. But it appears to me that in some references/papers it is assumed to be true.
