Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on $K^*_{orb}(\mathfrak{X}) \otimes \mathbb{C}$, the orbifold K-theory of any almost complex presentable orbifold $\mathfrak{X}$. We establish that under this stringy product, the delocalized Chern character

$$ch_{deloc} : K^*_{orb}(\mathfrak{X}) \otimes \mathbb{C} \longrightarrow H^*_{CR}(\mathfrak{X}),$$

after a canonical modification, is a ring isomorphism. Here $H^*_{CR}(\mathfrak{X})$ is the Chen-Ruan cohomology of $\mathfrak{X}$. The proof relies on an intrinsic description of the obstruction bundles in the construction of the Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryagin product (the latter is also called the fusion product in string theory).