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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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


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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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:
  • 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:
  • MathSciNet review: 4389792