# 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.
More »