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An interpretation of the Casselman-Wallach (C-W) Theorem is that the$K$-finite functor is an isomorphism of categories from the category offinitely generated, admissible smooth Fr\'echet modules of moderate growth tothe category of Harish-Chandra modules for a real reductive group, $G$ (here$K$ is a maximal compact subgroup of G).In this paper we study the dependenceof this functor on parameters. Our main result implies that holomorphicdependence implies holomorphic dependence. The work uses results from theexcellent thesis of van der Noort. Also a remarkable family of UniversalHarish-Chandra modules developed in this paper plays a key role.

The dependence on parameters of the inverse functor to the $K$-finite functor