Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact subgroup of $G'$. Denote by $\mathfrak{g}$ and $\mathfrak{g}'$ the complexified Lie algebras of $G$ and $G'$ respectively. Let $\theta$ be the Cartan involution of $G$ corresponding to $K$.

Suppose that $\pi$ is a nontrivial unitarizable simple $(\mathfrak{g}',K')$-module. Is there a common way to obtain an induced $(\mathfrak{g},K)$-module from $\pi$?

If $\mathfrak{g}'$ is a Levi subalgebra of $\mathfrak{g}$, one may use the Zuckerman functor composed with the produced functor to obtain a $(\mathfrak{g},K)$-module. But what if $\mathfrak{g}'$ is not supposed to be a Levi subalgebra of $\mathfrak{g}$?

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact subgroup of $G'$. Denote by $\mathfrak{g}$ and $\mathfrak{g}'$ the complexified Lie algebras of $G$ and $G'$ respectively. Let $\theta$ be the Cartan involution of $G$ corresponding to $K$.

Suppose that $\pi$ is a nontrivial unitarizable simple $(\mathfrak{g}',K')$-module. Is there a common way to obtain an induced $(\mathfrak{g},K)$-module from $\pi$?

If $\mathfrak{g}'$ is a Levi subalgebra of a $\theta$-stable parabolic subalgebra of $\mathfrak{g}$, one may use the Zuckerman functor composed with the produced functor to obtain a $(\mathfrak{g},K)$-module. But what if $\mathfrak{g}'$ is not supposed to be a Levi subalgebra of $\mathfrak{g}$?