Isomorphisms preserving invariants

Abstract

Let V and W be finite dimensional real vector spaces and let $${G \subset {\rm GL}(V)}$$ and $${H \subset {\rm GL}(W)}$$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to $${\mathbb{R}[V]^G}$$ and $${\mathbb{R}[W]^H}$$ , respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L −1 sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.