Equivariant K-theory of compact Lie groups with involution

For a compact simply connected simple Lie group $G$ with an involution $α$, we compute the $G\rtimes \Z/2$-equivariant K-theory of $G$ where $G$ acts by conjugation and $\Z/2$ acts either by $α$ or by $g\mapsto α(g)^{-1}$. We also give a representation-theoretic interpretation of those groups, as well as of $K_G(G)$.

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