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Representation Theory

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A proof of Casselman’s comparison theorem
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by Ning Li, Gang Liu and Jun Yu PDF
Represent. Theory 25 (2021), 994-1020 Request permission

Abstract:

Let $G$ be a real linear reductive group and $K$ be a maximal compact subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie algebra $\mathfrak {p}$, and $\mathfrak {n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V)$ ($i\geq 0$), where $V_{K}$ denotes the $(\mathfrak {g},K)$ module of $K$ finite vectors in $V$. This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any $k\geq 1$, $\mathfrak {n}^{k}V$ is a closed subspace of $V$ and the inclusion $V_{K}\subset V$ induces an isomorphism $V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V$. This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).
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Additional Information
  • Ning Li
  • Affiliation: Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Beijing 100871, People’s Republic of China
  • ORCID: 0000-0003-0775-9717
  • Email: lining@bicmr.pku.edu.cn
  • Gang Liu
  • Affiliation: Institut Elie Cartan de Lorraine, CNRS-UMR 7502, Université de Lorraine, 3 rue Augustin Fresnel, 57045 Metz, France
  • Email: gang.liu@univ-lorraine.fr
  • Jun Yu
  • Affiliation: Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Beijing 100871, People’s Republic of China
  • Email: junyu@bicmr.pku.edu.cn
  • Received by editor(s): May 6, 2021
  • Received by editor(s) in revised form: August 25, 2021, and August 29, 2021
  • Published electronically: December 1, 2021
  • Additional Notes: The third author’s research was partially supported by the NSFC Grant 11971036
    The third author is the corresponding author
  • © Copyright 2021 American Mathematical Society
  • Journal: Represent. Theory 25 (2021), 994-1020
  • MSC (2020): Primary 22E46
  • DOI: https://doi.org/10.1090/ert/591
  • MathSciNet review: 4346020