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.

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