Logarithmic trace and orbifold products

We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the “logarithmic trace” of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) taken with complex coefficients.