Functoriality and K-theory for $GL_n(\mathbb{R})$

TL;DR

This paper explores the K-theory of $GL_n(\mathbb{R})$ in relation to base change and automorphic induction, analyzing the Baum-Connes correspondence and computing the C*-algebra K-theory for this real reductive group.

Contribution

It provides detailed computations of the K-theory for $GL_n(\mathbb{R})$ and examines the interaction of base change with the Baum-Connes correspondence, extending previous work to the archimedean case.

Findings

Computed the C*-algebra K-theory of $GL_n(\mathbb{R})$

Analyzed the interaction of base change with Baum-Connes correspondence

Extended noncommutative geometry methods to real groups

Abstract

We investigate base change and automorphic induction $\mathbb{C}/\mathbb{R}$ at the level of K-theory for the general linear group $GL_n(\mathbb{R})$. In the course of this study, we compute in detail the C*-algebra K-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for $GL_n(\mathbb{R})$ and $GL_n(\mathbb{C})$. This article is the archimedean companion of our previous article in the Journal of Noncommutative Geometry.