Let $V=k^n$ for an algebraically closed field $k$ of characteristic 0, and let $W \subseteq V$ a subspace. Let $G\_W\subseteq GL(V)$ be the set of invertible linear maps that preserve $W$, i.e.
$$
G\_W=\{x \in GL(V): x(W)=W\},
$$
and let
$$
\mathfrak{g}\_W=\{X \in \mathfrak{gl}(V) : X(W) \subseteq W\},
$$
where $\mathfrak{gl}(V)$ is the Lie algebra of $GL(V)$, identified with the set of linear maps on $V$ under the commutator bracket. It is known (see e.g. Humphrey's Linear Algebraic Groups section 13.8) that $\mathfrak{g}\_W$ is the Lie algebra of $G\_W$.


My question is: If $Y \subseteq V$ is an arbitrary irreducible affine variety, what is the Lie algebra of its preserver $G\_Y$? I have a guess that would be consistent with the above result:


**Guess:** Suppose $Y \subseteq V$ is a homogeneous irreducible affine variety (i.e. $Y$ is a cone: $v \in Y \iff \alpha v \in Y$ for all $\alpha \in k$). Let


$$
\mathfrak{g}\_Y=\{X \in \mathfrak{gl}(V) : X(\mathscr{L}(Y)) \subseteq \mathscr{L}(Y)\},
$$
where $\mathscr{L}(Y) \subseteq V$ is the tangent space to $Y$ at $0$, i.e., if $Y=V(f\_1,\dots, f\_m)$, then $\mathscr{L}(Y)=V(d\_0f\_1,\dots, d\_0 f\_m)$, where $d\_0 f (x)= \sum\_{j=1}^n \frac{\delta f}{\delta x\_i} (0) x\_i$. Then $\mathfrak{g}\_Y$ is the Lie algebra of $G\_Y$.


Is this guess correct? If so, I would appreciate a proof or a reference to a proof.


A quick proof of the above result for when $Y=W$ is a subspace is provided by ShinyaSakai in the comments to [this question](https://math.stackexchange.com/q/51952/748775)


**EDIT**
@abx quickly disproved my guess, so my new question is simply: What is the Lie algebra of $G\_Y$? I would appreciate any relevant references in this vein.



The Lie algebra of the subgroup of $GL(n)$ preserving a given variety
Doc, you are right but only amorally. You need to replace the tangent vectors with jets to capture the behavior of your cone.


Let $I(Y)$ be the ideal of zeroes of your $Y$. Then 
$$
Lie (G\_Y) = \{ X \in {\mathfrak{gl}}(V) | X(I(Y))\subseteq I(Y)\}.
$$
Now you know that $I(Y)$ is homogeneous. Pick a finite set of its generators. Let $n$ be the highest degree of a generator from your set. Consider $n$-cojets 
$$
J^\ast(Y) := I(Y)/(I(Y)\cap I(0)^{n+1}) \subseteq J^\ast := I(0)/I(0)^{n+1}
$$
and $n$-jets
$$
J(Y) := J^{\ast}(Y)^\perp \subseteq J := (J^\ast)^\ast
$$
where $I(0)$ is the principal ideal.
This yields a desired "finite-dimensional" condition
$$
Lie (G\_Y) = \{ X \in {\mathfrak{gl}}(V) | X(J(Y))\subseteq J(Y)\}.
$$


