  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.
