  We prove inverse spectral results for differential operators on manifolds and
orbifolds invariant under a torus action. These inverse spectral results
involve the asymptotic equivariant spectrum, which is the spectrum itself
together with "very large" weights of the torus action on eigenspaces. More
precisely, we show that the asymptotic equivariant spectrum of the Laplace
operator of any toric metric on a generic toric orbifold determines the
equivariant biholomorphism class of the orbifold; we also show that the
asymptotic equivariant spectrum of a T^n-invariant Schrodinger operator on R^n
determines its potential in some suitably convex cases. In addition, we prove
that the asymptotic equivariant spectrum of an S^1-invariant metric on S^2
determines the metric itself in many cases. Finally, we obtain an asymptotic
equivariant inverse spectral result for weighted projective spaces. As a
crucial ingredient in these inverse results, we derive a surprisingly simple
formula for the asymptotic equivariant trace of a family of semi-classical
differential operators invariant under a torus action.
