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

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 $σ_G$ and the group inversion. We view $(G, a_G)$ as a Real $(G, σ_G)$-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.