Chern Classes and Compatible Power Operations in Inertial K-theory

Abstract

Let [X/G] be a smooth Deligne-Mumford quotient stack. In a previous paper the authors constructed a class of exotic products called inertial products on K(I[X/G]), the Grothendieck group of vector bundles on the inertia stack I[X/G]. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When G is diagonalizable these give rise to an augmented $\lambda$-ring structure on inertial K-theory. One well-known inertial product is the virtual product. Our results show that for toric Deligne-Mumford stacks there is a $\lambda$-ring structure on inertial K-theory. As an example, we compute the $\lambda$-ring structure on the virtual K-theory of the weighted projective lines P(1,2) and P(1,3). We prove that after tensoring with C, the augmentation completion of this $\lambda$-ring is isomorphic as a $\lambda$-ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles $T^*P(1,2)$ and $T^*P(1,3)$, respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the Hyper-Kaehler Resolution Conjecture.