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 corresponding 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, the orbifold products of Chen-Run, Abramovich-Graber-Vistoli, Jarvis-Kaufmann-Kimura, and Edidin-Jarvis-Kimura and the virtual product of Gonzalez-Lupercio-Segovia-Uribe-Xicotencatl.   We also introduce an entirely new product we call the localized orbifold product, which is defined on the complexification of K(IX).   The inertial products developed in this paper are used in a subsequent paper 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-Kaehler Resolution Conjecture. 
Dan Edidin, Tyler J. Jarvis, and Takashi Kimura
math.AG math.AT