Let $G$ be a (linearly reductive) algebraic group over $\mathbb{C}$ and let $V$ be a (rational) representation of $G$. Given a subspace $W\subseteq V$, let $N_G(W)=\{g\in G\mid gW=W\}$ denote its stabilizer. Is it true that the invariant ring $\mathbb{C}[W]^{N_G(W)}$ equals the restriction $\mathbb{C}[V]^G\mid_W$ of $\mathbb{C}[V]^G$ to $W$? If not, is it at least always true that $\mathbb{C}[W]^{N_G(W)}$ is finite over $\mathbb{C}[V]^G\mid_W?$
My thoughts: One direction is simple. $\mathbb{C}[V]^G\mid_W$ is necessarily contained in $\mathbb{C}[W]^{N_G(W)}$. I've no idea for the other direction. After trying a couple of things, I am convinced that the equality most probably does not hold but I was unable to produce a counter-example.
$\def\CC{\mathbb{C}}$They are not always equal. For example, let $\zeta$ be a cube root of unity, and let the cyclic group of order $3$ act on $V=\CC^2$ by $(x,y) \mapsto (\zeta x, \zeta^{-1} y)$. Then $\CC[V]^G = \CC[x^3, x y, y^3]$. Now let $W$ be any line other than the $x$ or $y$-axis. Then $N_G(W)$ is trivial, so $\CC[W]^{N_G(W)} = \CC[W]$ is a polynomial ring in one variable. But $\CC[V]^G$ has no elements in homgenous degree $1$, so $\CC[V]^G \to \CC[W]$ is not surjective.
Here is an example where $\CC[W]^{N_G(W)}$ is not finite over $\CC[V]^G$. Let $V$ be the vector space of $2 \times 3$ matrices, $\left[ \begin{smallmatrix} x_{11}&x_{12}&x_{13} \\ x_{21}&x_{22}&x_{23} \\ \end{smallmatrix} \right]$ and let $G$ be $SL_2$, acting by left multiplication. Then $\CC[V]^G$ is generated by the $2 \times 2$ minors, $x_{11} x_{22} - x_{21} x_{12}$, $x_{11} x_{23} - x_{21} x_{13}$ and $x_{12} x_{23} - x_{22} x_{13}$.
Let $W$ be the subspace $x_{21} = x_{13}=0$, in other words, matrices of the form $\left[ \begin{smallmatrix} \ast&\ast&0 \\ 0&\ast&\ast \\ \end{smallmatrix} \right]$. Then $N_G(W)$ is matrices of the form $\left[ \begin{smallmatrix} t&0 \\ 0&t^{-1} \end{smallmatrix} \right]$.
Then $$\CC[W]^{N_G(W)} = \CC[x_{11} x_{22},\ x_{11} x_{23},\ x_{12} x_{23},\ x_{12} x_{22}] \cong \CC[p,q,r,s]/\langle pr-qs \rangle$$ but the image of $\CC[V]$ is the subring $$\CC[x_{11} x_{22},\ x_{11} x_{23},\ x_{12} x_{23}] \cong \CC[p,q,r].$$
To see that $\CC[p,q,r,s]/\langle pr-qs \rangle$ is not finite over the subring $ \CC[p,q,r]$, pass to the quotient by $\langle p,q \rangle$. We $\CC[p,q,r,s]/\langle pr-qs,\ p,\ q \rangle \cong \CC[r,s]$ but $\CC[p,q,r]/\langle p,q \rangle \cong \CC[r]$, and $\CC[r,s]$ is clearly not finite over $\CC[r]$. $\square$