The ring structure for equivariant twisted K-theory

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map T(1) : H* (Gamma(center dot); A) -> H*(-1) ((N x Gamma)(center dot);A) for any crossed module N -> Gamma and prove that any element in the image is infinity-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N -> Gamma and any e is an element of Z(3) (Gamma(center dot);G(1)), that the equivariant twisted K-theory group K(e,Gamma)*(N) admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group K([e],G)*(G), defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure K([e],G)(i+d)(G) circle times K([e],G)(j+d)(G) -> K([e],G)(i+j+d)(G), where d = dim G and [c] is an element of H(2)((G x G)(center dot); S(1)). The relation with Freed-Hopkins-Teleman theorem [25] still needs to be explored.