Lie algebras of subspace stabilizers and Lie-stable subspaces

Given: An affine group scheme $G$ of finite type over $k$ with Lie algebra $\mathfrak{g}$ (The Lie algebra of a group scheme), a rational representation $(V,r)$ with differential $\mathrm{d}r:\mathfrak{g}\to {\mathfrak{g}\mathfrak{l}}_V$, a subspace $W\subseteq V$, and the scheme-theoretic stabilizer ${\mathrm{Stab}}_G(W)$ with $R$-points $\{g\in G(R):r(g)W_R=W_R\}$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers).

The universal stabilizer equations. Put $A=O(G)$ and $Q=V\mathrm{/}W$. The representation coaction and its inverse coaction are $\rho :V\to V\otimes A$ and $(\mathrm{id}\otimes S)\rho$. Under AC, $Q$ has a basis by Every vector space has a basis, so its coordinate functionals detect zero in $Q{\otimes }_{k}R$ for every $k$-algebra $R$. Applying these functionals to the images of $w\in W$ under both coactions gives coefficient equations in $A$ for $r_R(g)W_R\subseteq W_R$ and $r_R(g{)}^{-1}{W}_R\subseteq W_R$. Together they express equality and cut out a closed subgroup scheme of $G$. Its coordinate algebra is a quotient of $A$, hence finitely generated over $k$ by An affine scheme of finite type over a field has a finitely generated coordinate ring; thus the stabilizer is of finite type, even when $V$ is infinite-dimensional. (Rational representations and comodules of an affine group scheme, Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers, Fibre product of schemes)

The differential action. A dual-number point $e^{\epsilon X}$ acts on $V[\epsilon ]$ by $1+\epsilon \mathrm{d}r(X)$, with inverse $1-\epsilon \mathrm{d}r(X)$. This follows directly by evaluating the representation coaction at a point reducing to the identity. For finite-dimensional $V$, it is the matrix calculation $\mathrm{Lie}({\mathrm{GL}}_n)=M_n(k)$ of The Lie algebra of the general linear group; the same coaction calculation works for arbitrary $V$ without treating its automorphism functor as a finite-type scheme. (The Lie algebra of a group scheme, The tangent space at the identity is a vector space, and Lie is a functor)

Left exactness of $\mathrm{Lie}$. For algebraic subgroups $H_1,H_2\subseteq G$ with fibre product over a morphism, $\mathrm{Lie}$ commutes with the fibre product: in particular the Lie algebra of the pullback of a closed subgroup under a morphism is the fibre product of the Lie algebras, and $\mathrm{Lie}(H_1\cap H_2)=\mathrm{Lie}(H_1)\cap \mathrm{Lie}(H_2)$ (see (a)); moreover if $\mathrm{Lie}(H)=\mathrm{Lie}(G)$, $H$ is smooth and $G$ is connected, then $H=G$ (see (b)) (The Lie functor: exactness, fixed points and generation, The tangent space at the identity is a vector space, and Lie is a functor).

Cartier's theorem. In characteristic $0$ every affine group scheme of finite type over $k$ is smooth (Cartier's theorem: affine group schemes in characteristic zero are smooth; AC is used here and is inherited by this item).