Elliptic Operators Associated with Groups of Quantized Canonical Transformations

Abstract

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.

23 pages

References in corpus (1)

• Boundary Value Problems on Manifolds with Fibered Boundary

Cited by in corpus (2)

• Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations
• Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on ${\mathbb{R}}^n$