In Highlights 9 of this series, we showed that for an algebraic group $ G$ and a closed subgroup $ H\subseteq G$, we can always choose a representation $ G\to\rm{GL}(V)$ with a line $ L\subseteq V$ whose stabiliser is $ H$. In turn, this allows us to identify the quotient $ G/H$ with the orbit of the class $ [L]$ in the projective space $ \mathbf{P}(V)$, which satisfies the universal property for quotients, thereby giving us a sensible variety structure on $ G/H$.

In this post, we specialise to the case of a Borel subgroup $ B\leq G$; that is $ B$ is maximal amongst the connected solvable groups. Such a subgroup is necessarily closed!

The fact that will allow us to study Borel subgroups is the fixed point theorem: a connected solvable group that acts on a nonempty complete variety has a fixed point. By choosing a representation $ G\to \rm{GL}(V)$ with a line $ L\subseteq V$ whose stabiliser is $ B$, we get identify $ G/B$ with a quasiprojective variety. However, in this case $ G/B$ is actually projective. Here is a short sketch:
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