$KR$-theory of compact Lie groups with group anti-involutions

TL;DR

This paper computes the equivariant $KR$-theory ring structure of certain compact Lie groups with anti-involutions, generalizing known results from complex $K$-theory to the real $KR$-theory setting.

Contribution

It introduces the notion of Real equivariant formality to explicitly determine the ring structure of equivariant $KR$-theory for Lie groups with anti-involutions, extending Brylinski-Zhang's results.

Findings

The equivariant $KR$-theory ring is the Grothendieck differentials of the coefficient ring.

The computation applies when the group has no Real representations of complex type.

Generalizes Brylinski-Zhang's results from complex $K$-theory to $KR$-theory.

Abstract

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with an anti-involution $a_G$ which is the composition of a Lie group involutive automorphism $\sigma_G$ and the group inversion. We view $(G, a_G)$ as a Real $(G, \sigma_G)$-space via the conjugation action. In this note, we exploit the notion of Real equivariant formality discussed in \cite{Fo} to compute the ring structure of the equivariant $KR$-theory of $G$. In particular, we show that when $G$ does not have Real representations of complex type, the equivariant $KR$-theory is the ring of Grothendieck differentials of the coefficient ring of equivariant $KR$-theory over the coefficient ring of ordinary $KR$-theory, thereby generalizing a result of Brylinski-Zhang's (\cite{BZ}) for the complex $K$-theory case.