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.
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