# Dependence on parameters of CW globalizations of families of Harish-Chandra modules and the meromorphic continuation of $C^{\infty}$ Eisenstein series

## TL;DR

This paper proves the parameter dependence of CW globalization for families of Harish-Chandra modules and establishes the meromorphic continuation of $C^{
abla}$ Eisenstein series, advancing understanding in representation theory and automorphic forms.

## Contribution

It demonstrates the continuity of CW globalization in parameter families and proves the meromorphic continuation of $C^{
abla}$ Eisenstein series using Langlands' results.

## Key findings

- CW globalization depends continuously on parameters.
- Meromorphic continuation of $C^{
abla}$ Eisenstein series established.
- Application of Langlands' results in the $K$-finite case.

## Abstract

The first main result is that the Casselman-Wallach Globalization of a real analytic family of Harish-Chandra modules is continuous in the parameter. Our proof of this result uses results from the thesis of Vincent van der Noort in several critical ways. In his thesis the holomorphic version of the result was proved in the case when the parameter space is a one dimensional complex manifold up to a branched covering. The second main result is a proof of the meromorphic continuation of $C^{\infty}$ Eisenstein series.using Langlands' results in the $K$ finite case as an application of the methods in the proof of the first part.