# Highlights in Linear Algebraic Groups 9: Quotients as Varieties

Conventions: an algebraic group here is a linear algebraic group over a fixed algebraically closed field $k$.

In Highlights 6 and Highlights 7 in this series on algebraic groups, we saw that given any finite dimensional $k$-subspace $F\subseteq k[G]$, we can find a finite dimensional subspace $E\supseteq F$ such that $E$ is stable under all left-translations. (The same method shows that we can choose $E$ invariant under all right translations in the case of $G\times G\to G$. Explicitly, if $f\in k[G]$ and $x\in G$ then the right translation $\rho_x$ is defined by $(\rho_xf)(y) = f(yx)$.)

Using either left or right translations, we can use the techniques we have seen to construct for any closed subgroup $H\subseteq G$ a representation $\phi:G\to \rm{GL}(V)$ that contains a line $L$ whose stabiliser in $G$ is exactly $H$. This is an extremely important technique that will allow us to realise the homogeneous space $G/H$ as a quasiprojective variety! This will come in handy later for studying Borel subgroups $B\subseteq G$. In this case we will see that $G/B$ is actually projective (and in our setting, equivalently, complete), which is a key observation for deducing the structure of reductive groups.

In this post, we shall go through the construction of this representation.

### The Recipe

Ingredients:

• An algebraic group $G$
• A closed subgroup $H\subseteq G$

Here we explain how to construct the representation $\phi:G\to\rm{GL}(V)$ such that $V$ contains a line stabilised by $H$.

We let $I\subseteq k[G]$ be the ideal defining $H$, and $\{ f_1,\dots,f_n\}$ be a finite set of generators. The $k$-span $F$ of these generators is a finite dimensional space $F$, and we can choose a finite-dimensional subspace $E\supseteq F$ invariant under all left translations.

Let $M = E\cap I$. Clearly, $M$ contains a generating set for $I$, and is invariant under all left translations $\lambda_x$ for $x\in H$. This is because $E$ is invariant under all $\lambda_x$ for $x\in G$ and $I$ is invariant under all $\lambda_x$ for $x\in H$ (in fact, a short calculation shows that $H$ can be characterised as the set $\{ x\in G : \lambda_xI\subseteq I\}$).

It is also an exercise to verify that the stabiliser of $M$ is exactly $H$ (use the fact that $M$ generates $I$ and is stable under left translations $\lambda_x$ for $x\in H$).

Let $d = \dim_k(M)$. Elementary linear algebra shows that $L = \wedge^d M$ has stabiliser $H$ in the representation $\wedge^d E$.

### Projective Space

Now $G$ acts on $V$, and $L\subseteq V$ is a line whose stabiliser is $H$. Then $G$ also acts on $\mathbf{P}(V)$, and in $\mathbf{P}(V)$, the line $L$ becomes the point $[L]$. We have an orbit morphism $G\to \mathbf{P}(V)$ defined by $g\mapsto [\phi(g)L]$. We get a set-theoretic bijection $G/H\to\mathbf{P}(V)$, which identifies $G/H$ with the quasiprojective variety that is the orbit of the point $[L]$. It is not hard to verify that the orbit of $[L]$ satisfies a universal property for quotients; since we will not need this, let us just remark that when we speak of $G/H$ as a variety, we mean the orbit of $[L]$ under the action of $G$; the universal property shows that this is well-defined.