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

Let G   be a compact, connected, and simply-connected Lie group, equipped with an anti-involution aGaG which is the composition of a Lie group involutive automorphism σGσG and the group inversion. We view (G,aG)(G,aG) as a Real (G,σG)(G,σG)-space via the conjugation action. In this note, we exploit the notion of Real equivariant formality discussed in [9] 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 [8] for the complex K-theory case.