We prove an inverse spectral result for S1-invariant metrics on S2
based on the so-called asymptotic equivariant spectrum. This is roughly the
spectrum together with large weights of the S1 action on the eigenspaces.
Our result generalizes an inverse spectral result of the first and last named
authors, together with Victor Guillemin, concerning S1-invariant metrics on
S2 which are invariant under the antipodal map. We use higher order terms in
the asymptotic expansion of a natural spectral measure associated with the
Laplacian and the S1 action.Comment: 16 pages; minor revisions throughout following comments from referee