Linearization of proper groupoids

TL;DR

This paper proves that smooth proper Lie groupoids near a fixed point can be locally linearized, showing they are equivalent to linear actions of compact Lie groups, which advances understanding of their local structure.

Contribution

It establishes the local linearizability of proper Lie groupoids near fixed points, confirming a conjecture by Weinstein and extending linearization results to neighborhoods of orbits.

Findings

Proper Lie groupoids are locally linearizable near fixed points.

The result confirms Weinstein's conjecture on local linearization.

Linearization applies in neighborhoods of orbits under mild conditions.

Abstract

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth linearizability of a proper Lie groupoid in the neighborhood of an orbit under a mild condition.