Equivariant K-Theory of Simply Connected Lie Groups

We compute the equivariant $K$-theory $K_G^{\ast }(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^{\ast }(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3), and compute the corresponding equivariant $K$-theory.