Homotopy principles for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\text{\hspace{0.17em}⁣}:X\to Q_X$ and $p_Y\text{\hspace{0.17em}⁣}:Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $\varphi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\varphi$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:X\to Y$, inducing $\varphi$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $\mathrm{\Phi }$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(\varphi (q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $\mathrm{\Phi }$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.