The Real K-Theory of Compact Lie Groups

TL;DR

This paper characterizes the ring structure of equivariant KR-theory for compact, simply-connected Lie groups with involution, extending previous work on module and ring structures in K-theory.

Contribution

It provides a detailed description of the ring structure of equivariant KR-theory for such Lie groups, integrating prior results on module and ring structures.

Findings

Explicit description of the ring structure of equivariant KR-theory.

Connection between KR-theory and previous K-theory results.

Enhanced understanding of involution effects on Lie group K-theories.

Abstract

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution $\sigma_G$ and viewed as a $G$-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) $KR$-theory of $(G, \sigma_G)$ by drawing on previous results on the module structure of the $KR$-theory and the ring structure of the equivariant $K$-theory.