Lower order asymptotics for Szeg\"{o} and Toeplitz kernels under Hamiltonian circle actions

We consider a natural variant of Berezin-Toeplitz quantization of compact K"{a}hler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szeg"{o} and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work on near-diagonal scaling asymptotics of equivariant Szeg"{o} kernels in the presence of Hamiltonian torus actions.