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.