K-twisted K-theory for SU(N)

We present a version of twisted equivariant $K$-theory-$K$-twisted equivariant $K$-theory, and use Grothendieck differentials to compute the $K$ -twisted equivariant $K$-theory of simple simply connected Lie groups. We did the calculation explicitly for SU(N) explicitly. The basic idea is to interpret an equivariant gerbe as an element of equivariant $K$-theory of degree 1.