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Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for some reductive subgroup $K$ of $G$.

In this case we obtain an action by vector fields of the Lie algebra $\mathfrak{g}$ of $G$ on functions on $X$, the algebra $k[X]$. Choose an element $u\in\mathfrak{g}$.

My question is: are there statements about nice local forms of $u$? e.g., I would like to know about statements of the form: If $u$ vanishes at a point $a$ on $X$ there exists a neighborhood $U$ of $a$ with coordinates $x_1,\dots,x_n$ (coming from an etale map $U\to\mathbb{A}^n$) such that $$u=\sum\limits_i x_{j_i}\partial_{x_{k_i}}$$ Any references or comments are greatly appreciated!

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Let's look on the linearization of the action (it exists for a lie group $G$ acting on affine variety $X$). That way, $X$ is embedded into an affine space $V = \mathbb{A}^N$ as a closed subset, and $G$ acts linearly on $V$. Consider now an element $s \in \mathfrak{g}$ and the corresponding linear vector field on $V$, let $x \in X$ be its fixed point. $T_X(x)$ is an invariant subspace of $s$. Provided it would have an invariant complement, projection along this complement would give us the desired form.

For reductive $K$ we can do even better - $Stab_x \cong K$, and $K$ is reductive, so we can chose the invariant complement with respect to whole $K$. Projection along this complement on $T_x(X)$ gives the desired etale linearization for the whole stabilizer simultaneously.

EDIT: actually I've realised that you ask about somewhat strange normal form. I read it automatically as linear vector field ($\Sigma a_{ij}x^i \partial_j$). The rest is linear algebra anyways (the form you are demanding as it is written I think does not exist even for linear vector fields).

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