Invariant Whitney Functions

TL;DR

This paper extends Schwarz's theorem to Whitney functions along certain subanalytic sets, using groupoids to handle non-G-stable cases, thereby broadening the understanding of invariant functions in geometric analysis.

Contribution

It proves a new invariance theorem for Whitney functions on subanalytic sets, generalizing Schwarz's result to non-G-stable sets via groupoid techniques.

Findings

Invariant Whitney functions can be expressed as compositions with a generalized Hilbert map.

The approach handles non-G-stable sets using the language of groupoids.

The theorem broadens the class of sets where invariant function representation is possible.

Abstract

A theorem of Gerald Schwarz [24, Thm. 1] says that for a linear action of a compact Lie group $G$ on a finite dimensional real vector space $V$ any smooth $G$-invariant function on $V$ can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set $Z\subset V$ fulfilling some regularity assumptions. In order to deal with the case when $Z$ is not $G$-stable we use the language of groupoids.