# Induced $(\mathfrak{g},K)$-modules

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 $$\mathfrak{g}$$?