Smooth Fréchet globalizations of Harish-Chandra modules (Q2017110)

In the paper, essentially self-contained and comprehensive, accessible to graduate students, the authors deal with Harish-Chandra modules, that is, finitely generated \((\mathfrak{g},K)\)-modules with finite \(K\)-multiplicities, where \(K\) is a maximal compact subgroup of a linear reductive real Lie group \(G\) with Lie algebra \(\mathfrak{g}.\) They define a \textit{globalization} of a Harish-Chandra module \(V\) as a representation \((\pi,E)\) of \(G\) such that the \(K\)-finite vectors of \(E\) are isomorphic to \(V\) as a \((\mathfrak{g},K)\)-module and denote by \textit{SAF} the category whose objects are smooth admissible moderate growth Fréchet representations of \(G\) with continuous linear \(G\)-maps as morphisms and they use it to present a new approach to the Casselman-Wallach globalization theorem by giving an alternative proof of it based on lower bounds for matrix coefficients on a reductive group. To do this, they introduce the concept of \textit{good} Harish-Chandra module if it admits a unique \textit{SAF}-globalization and they prove that this quality of a Harish-Chandra module is preserved by extensions, induction and tensoring with finite-dimensional representations.