Fr\'echet completions of moderate growth old and (somewhat) new results

TL;DR

This paper provides a new proof of the meromorphic continuation of smooth Eisenstein series by clarifying the Casselman-Wallach theorem and combining it with a theorem of van der Noort, offering insights into the structure of representations of real reductive groups.

Contribution

It introduces a novel proof of the meromorphic continuation of Eisenstein series using an improved understanding of the Casselman-Wallach theorem and related representation theory results.

Findings

New proof of meromorphic continuation of Eisenstein series

Clarification of the Casselman-Wallach theorem

Integration of van der Noort's theorem with existing proofs

Abstract

This article has two objectives. The first is to give a guide to the proof of the (so-called) Casselman-Wallach theorem as it appears in Real Reductive Groups II. The emphasis will be on one aspect of the original proof that leads to the new result in this paper which is the second objective. We show how a theorem of van der Noort combined with a clarification of the original argument in my book lead to a theorem with parameters (an alternative is one announced by Berstein and Kr\"otz). This result gives a new proof of the meromorphic continulation of the smooth Eisenstein series.