Index of nonlocal elliptic boundary value problems associated with isometric group actions

TL;DR

This paper establishes an index theorem for nonlocal elliptic boundary value problems on manifolds with boundary, incorporating group actions and pseudodifferential operators, extending classical index theory to nonlocal settings.

Contribution

It introduces an index theorem for elliptic elements in a nonlocal operator algebra generated by boundary value problems and group shift operators on manifolds with boundary.

Findings

Formulation of an index theorem for nonlocal elliptic operators

Extension of Boutet de Monvel algebra to include group actions

Application to manifolds with boundary and isometric group actions

Abstract

Given a compact manifold with boundary endowed with an isometric action of a discrete group of polynomial growth, we state an index theorem for elliptic elements in the algebra of nonlocal operators generated by the Boutet de Monvel algebra of pseudodifferential boundary value problems on the manifold and the shift operators associated with the group action.