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

Wallach, Nolan

doi:10.1090/ert/596

The dependence on parameters of the inverse functor to the $K$-finite functor
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Represent. Theory 26 (2022), 94-121 Request permission

Abstract:

An interpretation of the Casselman-Wallach Theorem is that the $K$-finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the 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 dependence of the inverse functor to the $K$-finite functor on parameters. Our main result implies that holomorphic dependence implies holomorphic dependence. The work uses results from the excellent thesis of van der Noort. Also a remarkable family of universal Harish-Chandra modules, developed in this paper, plays a key role.

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