  In this paper, we provide a simpler proof of the Kr\"otz--Stanton Extension
Theorem for holomorphic extensions of orbit maps in unitary irreducible
representations of connected semisimple Lie groups. Our result is a
generalization of the previous result as it applies to Hilbert globalizations
of finitely generated admissible $(\mathfrak{g},K)$-modules for arbitrary
connected semisimple Lie groups $G$ with not necessarily finite center.
Furthermore, we give polynomial growth estimates for the norms of the
holomorphic extensions of the orbit maps of $K$-finite vectors in irreducible
unitary representations. We use this to prove that, at the boundary of the
principal crown bundle, these holomorphic extensions have boundary values in
the space of distribution vectors.
