The dependence on parameters of the inverse functor to the $K$-finite functor
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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.References
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Additional Information
- Nolan R. Wallach
- Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California
- MR Author ID: 180225
- ORCID: 0000-0002-0656-2421
- Email: nwallach@ucsd.edu
- Received by editor(s): February 5, 2021
- Received by editor(s) in revised form: August 3, 2021, August 24, 2021, and September 10, 2021
- Published electronically: March 4, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory 26 (2022), 94-121
- MSC (2020): Primary 22E45, 22E30
- DOI: https://doi.org/10.1090/ert/596
- MathSciNet review: 4389792