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Given a Lie group G of quantized canonical transformations acting on the space L2(M) over a closed manifold M , we define an algebra of so-called G-operators on L2(M). We show that to G-operators we can associate symbols in appropriate crossed products with G, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions onM , transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.

Elliptic operators associated with groups of quantized canonical transformations