Elliptic operators associated with groups of quantized canonical transformations

Given a Lie group G of quantized canonical transformations acting on the space L-2(M) over a closed manifold M, we define an algebra of so-called G-operators on L-2(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 on M, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds. (C) 2019 Published by Elsevier Masson SAS.