TITLE: Linearisation of a group
QUESTION [5 upvotes]: If $G$ is a connected  Lie group  acting on a vector $\mathbb{C}$-space $V$  then it is well known that the algebra of invariants  $\mathbb{C}[V]^G$ coincides with  the algebra of invariants $\mathbb{C}[V]^L$ of  corresponding Lie algebra $L.$
Question. Let now $G$ be a finite group, $V$ be its representation. Is there exists a Lie algebra $L$ and its representation on $V$ such that $\mathbb{C}[V]^G=\mathbb{C}[V]^L$?
It is easy to see  that it impossible  for symmetric group $S_n$. But maybe there are classes of finite groups for which can be found the positive answer? 
Edit. Let $V=< v_1,v_2,\ldots,v_n >$  be standard representation of the symmetric group $S_n.$ Suppose that there exist a derivation $D=f_1 \frac{\partial}{\partial x_1}+\cdots+ f_n \frac{\partial}{\partial x_n}, f_i \in S(V)$  of the symmetric algebra $S(V)$ such that $D(I)=0$ for all $I \in \mathbb{C}[V]^{S_n}$.
Let $I_1,I_2,\ldots, I_n \in S(V)$ be a minimal generating set for $\mathbb{C}[V]^{S_n}.$  Since $D(I_k)=0, \forall k,$  we get  the system of polynomial equations on $f_1,f_2,\ldots,f_n:$
$$
f_1 \frac{\partial I_1}{\partial x_1}+\cdots+ f_n \frac{\partial I_1}{\partial x_n}=0,\\
f_1 \frac{\partial I_2}{\partial x_1}+\cdots+ f_n \frac{\partial I_2}{\partial x_n}=0,\\
\ldots \\
f_1 \frac{\partial I_n}{\partial x_1}+\cdots+ f_n \frac{\partial I_n}{\partial x_n}=0.
$$
The Jacobian of system of invariants $I_1,I_2,\ldots, I_n$ is not zero. It follows   the  system has only trivial  solution $f_i=0.$

REPLY [3 votes]: You asked for it! Here's the Hilbert series argument. I need to assume that the derivation $D$ is homogeneous, or equivalently comes from extending a linear map $V \to V$. This covers the case of a Lie algebra action on $V$. 
First, recall that $\dim S^n(V^{\ast}) = {\dim V + n - 1 \choose \dim V - 1}$ has asymptotic growth rate $\Theta(n^{\dim V - 1})$ for $\dim V$ fixed and $n \to \infty$. 
Let $g_n = \dim S^n(V^{\ast})^G$ denote the dimension of the $n^{th}$ graded component of $\mathbb{C}[V]^G$. An explicit expression for the Hilbert series $\sum g_n t^n$ can be obtained from Molien's theorem which shows in particular that the leading term in the asymptotic expansion of $g_n$ is at least $\frac{1}{|G|} n^{\dim V - 1}$ (hence, as above, that $\mathbb{C}[V]^G$ has Krull dimension $\dim V$). There should be a more elementary way to obtain this estimate. 
On the other hand, let $D$ be a derivation induced from a linear map $V \to V$. Let $e_1, ..., e_n$ be a Jordan basis of $D$ with generalized eigenvalues $\lambda_1, ..., \lambda_n$. Then the monomial $e_1^{k_1} ... e_n^{k_n}$ in $\mathbb{C}[V]$ is a generalized eigenvector of $D$ with generalized eigenvalue $\sum k_i \lambda_i$. In particular, the $n^{th}$ graded component of $\mathbb{C}[V]^D$ has dimension at most the number of monomials of degree $n$ of generalized eigenvalue $0$. The exponents of these monomials are determined by the kernel of the map
$$\mathbb{Z}^n \ni (k_1, ..., k_n) \mapsto \sum_{i=1}^n k_i \lambda_i \in \mathbb{C}.$$
If at least one of the $\lambda_i$ is nonzero, the kernel of the above map is a submodule of rank at most $n-1$, from which it follows that the dimension $d_n$ of the $n^{th}$ graded component of $\mathbb{C}[V]^D$ has asymptotic growth rate at most as fast as $O(n^{\dim V - 2})$. 
If all of the $\lambda_i$ are equal to zero, then $D$ is nilpotent. In this case we need to inspect the action of $D$ on monomials more carefully. Permute the Jordan basis so that $D(e_i)$ is either $e_{i-1}$ or $0$ and so that $D(e_2) = e_1$, and consider the lexicographic order on $\mathbb{C}[V]$ with respect to the $e_i$. Then $D(e_1^{m_1} ... e_k^{m_k})$ has leading term $m_2 e_1^{m_1+1} e_2^{m_2-1} ... e_k^{m_k}$, from which it follows that the image of $D$ acting on $S^n(V^{\ast})$ has dimension at least the number of monomials with $m_1 \ge 1$, or the dimension of $S^{n-1}(V^{\ast})$. The two dimensions are polynomials in $n$ with the same leading term and $d_n$ is their difference, so we again find that the asymptotic growth rate of $d_n$ is at most $O(n^{\dim V - 2})$.