The purpose of this article is to give a purely equivariant definition of orbifold Chow rings of quotient Deligne-Mumford stacks. This completes a program begun in [JKK] for quotients by finite groups. The key to our construction is the definition (Section 6.1) of a twisted pullback in equivariant $K$ -theory, $K_G(X) \to K_G(\mathbb{I}_{G}^{2}(X))$ taking nonnegative elements to nonnegative elements. (Here $\mathbb{I}_{G}^{2}(X) = \{(g_1,g_2,x)|g_1x = g_2 x = x \} \subset G \times G \times X$ .) The twisted pullback is defined using data about fixed loci of elements of finite order in $G$ but depends only on the underlying quotient stack (Theorem 6.3). In our theory, the twisted pullback of the class ${\mathbb T} \in K_G(X)$ , corresponding to the tangent bundle to $[X/G]$ , replaces the obstruction bundle of the corresponding moduli space of twisted stable maps. When $G$ is finite, the twisted pullback of the tangent bundle agrees with the class $R({\mathbf m})$ given in [JKK, Definition 1.5]. However, unlike in [JKK] we need not compare our class to the class of the obstruction bundle of Fantechi and Göttsche [FG] in order to prove that it is a nonnegative integral element of $K_G(\mathbb{I}_{G}^{2}(X))$ .
¶ We also give an equivariant description of the product on the orbifold $K$ -theory of $[X/G]$ . Our orbifold Riemann-Roch theorem (Theorem 7.3) states that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we show (see Theorem 8.7) that if ${\mathscr X} = [X/G]$ , then there is an associative orbifold product structure on $K({\mathscr X})\otimes {\mathbb C}$ distinct from the usual tensor product