Restriction of holomorphic functions on $G$-invariant subspace Let $X$ be a complex manifold with a holomorphic action of a complex reductive group $G$. Let $Y \subset X$ be a $G$-invariant reduced complex analytic subspace. Is the restriction
$$
\mathcal{O}_X^G \to \mathcal{O}_Y^G
$$
surjective (on stalks)? Here $\mathcal{O}_X^G$ is the sheaf of $G$-invariant holomorphic functions on $X$ and similarly for $Y$.
 A: After writing an answer in terms of global sections, then realising the original question was in terms of stalks, I realised that I don't understand what "the sheaf of $G$-invariant holomorphic functions on $X$" means:  what is $\mathscr O_X^G(U)$, if $U$ is a non-$G$-stable open subset of $X$?  I wonder if you might mean the sheaf of holomorphic functions on the quotient $X/G$, which is defined by putting $\mathscr O_{X/G}(\overline U_X) = \mathscr O_X(U_X)^G$ for any open subset $\overline U_X$ of $X/G$ with pullback $U_X \subseteq X$.  I'll answer the question that way, but, since the technique I describe is very robust, hopefully I can adapt it to whatever the correct interpretation is.
Fix a point $\overline y \in Y/G$, and a germ $f$ at $\overline y$ (that is, an equivalence class of $G$-invariant holomorphic functions on $Y$ defined on the pullback of an open neighbourhood in $Y/G$ of $\overline y$).
Fix an open subset $\overline U_X$ of $X/G$ containing $\overline y$ that is small enough that $f$ is defined on $\overline U_X \cap Y/G$ and that, if we write $U_X$ for the pullback to $X$ of $\overline U_X$, then $f$ extends to a holomorphic function on $U_X$.  Then consider the space $\mathscr V$ of functions on $X$ whose restriction to $Y$ is proportional to $f$.  This is a $G$-representation, hence also a $\mathfrak g$-representation.
Let $K$ be the compact form of $G$.  Since $K$ is compact, the representation $\mathscr V$ has a non-$0$, $K$-fixed vector $F$.  The span of $F$ is annihilated by $\mathfrak k$, hence by $\mathfrak g = \mathfrak k \otimes_{\mathbb R} \mathbb C$; so $F$ itself is $G$-fixed.
