# Is every Zariski closed subgroup a stabilizer?

Let $$G$$ be a linear algebraic group. Is it true that a subgroup $$H$$ of $$G$$ is Zariski closed if and only if there exists a representation $$\pi: G \to \mathrm{GL}(V)$$ and a vector $$v \in V$$ such that the stabilizer $$G_v:=\{g \in G: \pi(g)v=v \}$$ is equal to $$H$$?

I think one implication is clear since $$G_v$$ is certainly a subgroup and the equation $$\pi(g)v=v$$ is polynomial in the matrix entries. Thus any stabilizer must be Zariski closed.

I am not sure of the reverse implication. Is it really true that every Zariski closed subgroup of $$G$$ arises as the stabilizer of some vector in some representation?

• One could still ask the question: when does a Zariski-closed subgroup $H$ appear as point stabilizer in one representation? This is true if $H$ has no nontrivial multiplicative character. Note that $H$ is not assumed connected and the question does not a priori boil down to this case (e.g., I don't know if when $H$ is finite, $H$ can always appear as a point stabilizer).
– YCor
Nov 28 '21 at 15:19

Let $$G$$ be a linear algebraic group and let $$H$$ be a Zariski closed subgroup. Then there is a representation $$V$$ of $$G$$ and a one dimensional subspace $$L$$ of $$V$$ such that $$H$$ is the stabilizer of $$\mathbb{P}(L)$$ in the action of $$G$$ on $$\mathbb{P}(V)$$.
To see that you can't get exactly what you asked for, work over a field of characteristic zero, take $$G = \text{SL}_2$$ and let $$H$$ be the Borel subgroup $$B:=\left[ \begin{smallmatrix} \ast & \ast \\ 0 & \ast \end{smallmatrix} \right]$$. The classification of $$\text{SL}_2$$-representations is well known, and we see that $$V^B = V^{\text{SL}_2}$$ for any $$\text{SL}_2$$-representation $$V$$.
• Is there any way to parlay this fact into an example of a homogeneous space $G/H$ which cannot be a linear group orbit $G/G_v$ for any representation? Nov 27 '21 at 14:46
• Yes, exactly. $SL_2/B$ is $\mathbb{P}^1$, which has no nonconstant global functions, hence cannot be embedded into any vector space. Nov 27 '21 at 14:57