Isomorphisms preserving invariants

TL;DR

This paper investigates the relationship between vector space isomorphisms that preserve group orbit structures and the induced algebraic variety isomorphisms, providing new proofs for related theorems in invariant theory.

Contribution

It establishes a correspondence between quasi-isomorphisms of vector spaces with group actions and isomorphisms of their invariant algebraic varieties, offering new proofs of existing theorems.

Findings

Quasi-isomorphisms induce isomorphisms of invariant varieties.

Diffeomorphisms of quotient spaces imply quasi-isomorphisms.

Provides new proofs of Strub's theorem and lifting of biholomorphisms.

Abstract

Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset\GL(V)$ and $H\subset\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\R[V]^G$ and $\R[W]^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\colon V\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\inv$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\colon Y\to Z$ 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 \cite{Strub}, for which we give a new proof. We also give a new proof of a result of \cite{KrieglLosikMichor03} on lifting of biholomorphisms of quotient spaces.