A Plethora of Inertial Products

For a smooth Deligne-Mumford stack X , we describe a large number of inertial products on K(IX ) and A∗(IX ) and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on K(IX ) and an inertial product on A∗(IX ) and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle V on X defines two new inertial pairs. We recover, as special cases, both the orbifold products of [CR04, AGV02, FG03, JKK07, EJK10] and the virtual product of [GLS+07]. We also introduce an entirely new product we call the localized orbifold product, which is defined on K(IX ) ⊗ C. The inertial products developed in this paper are used in a subsequent paper [EJK12] to describe a theory of inertial Chern classes and power operations in inertial K-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the Hyper-Kähler Resolution Conjecture.