Chern Classes and Compatible Power Operations in Inertial K-theory

We develop a theory of Chern classes and compatible power operations for strongly Gorenstein inertial pairs (in the sense of [EJK12]). An important application is to show that there is a theory of Chern classes and compatible power operations for the virtual product defined by [GLSUX07]. We also show that when the inertia stack IX is a quotient of the form IX = [X/G], with G diagonalizable, then inertial K-theory of IX has a lambda-ring structure. This implies that for toric Deligne-Mumford stacks there is a corresponding lambda-ring structure associated to virtual K-theory. As an example we compute the semi-group of lambda-positive elements in the virtual lambda-ring of the weighted projective stacks P(1,2) and P(1,3). Using the virtual orbifold line elements in this semi-group, we obtain a simple presentation of the K-theory ring with the virtual product and a simple description of the virtual first Chern classes. This allows us to prove that the completion of this ring with respect to the augmentation ideal is isomorphic to the usual K-theory of the resolution of singularities of the cotangent bundle T^* P(1,2) and T^*P(1,3), respectively. We interpret this as a manifestation of mirror symmetry, in the spirit of the Hyper-Kaehler Resolution Conjecture.