Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

TL;DR

This paper establishes a connection between the representation theories of Lie groupoids and infinite-dimensional Lie groups, showing how smooth representations translate between these structures and under what conditions this correspondence is continuous or smooth.

Contribution

It extends the known correspondence between Lie groupoid and Lie group representations from the topological to the smooth category under weaker assumptions.

Findings

Smooth representations of Lie groupoids induce representations of associated Lie groups.

Representations of Lie groupoids lead to continuous and smooth representations on bundle sections.

The correspondence between these representations can be extended to the smooth category with weaker assumptions.

Abstract

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids.