Functional equations for orbifold wreath products

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for the Euler and Euler--Satake characteristics, which we compute for a class of weighted projective spaces. This generalizes results known for global quotients by finite groups to all closed, effective orbifolds. We also describe a combinatorial approach to extensions of multiplicative invariants using decomposable functors that recovers the formula for the Euler--Satake characteristic of a wreath product of a global quotient orbifold.