Elliptic Operators Associated with Groups of Quantized Canonical Transformations

TL;DR

This paper develops a unified algebraic framework for elliptic operators linked to groups of quantized canonical transformations, generalizing many known elliptic theories and establishing their Fredholm properties.

Contribution

It introduces a new algebra of G-operators on L^2(M), defines symbols in crossed products with G, and proves ellipticity and Fredholm properties within this framework.

Findings

Unified framework for various elliptic theories

Established ellipticity and Fredholm properties for G-operators

Generalized shift, transversal, and Fourier integral operators

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.