By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.
"Melrose-Uhlmann Projectors, the Metaplectic Representation and Symplectic Cuts." J. Differential Geom. 61 (3) 365 - 396, July, 2002. https://doi.org/10.4310/jdg/1090351528