On the linearization theorem for proper Lie groupoids

TL;DR

This paper revisits and clarifies the linearization theorems for proper Lie groupoids, providing shorter proofs and a more conceptual understanding of the conditions needed for the theorems to hold.

Contribution

It offers a shorter, geometric proof of Zung's theorem and a conceptual interpretation of Weinstein's general orbit linearization, clarifying the necessary conditions.

Findings

Shorter, geometric proof of Zung's theorem

Conceptual interpretation of Weinstein's orbit linearization

Clarification of conditions for the theorems

Abstract

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).