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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中文翻译：

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逆函子的参数对 𝐾 有限函子的依赖性

摘要：对 Casselman-Wallach 定理的一种解释是，$K$-有限函子是从有限生成的、可容许的平滑 Fréchet 适度增长模到真正还原的 Harish-Chandra 模的类别的同构。群，$G$（这里 $K$ 是 $G$ 的最大紧子群）。在本文中，我们研究了逆函子对$K$-有限函子对参数的依赖性。我们的主要结果意味着全纯依赖意味着全纯依赖。这项工作使用了范德诺特出色论文的结果。在本文中开发的一个非凡的通用 Harish-Chandra 模块系列也起着关键作用。

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