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:

Sketch

We start by choosing $ B$ specifically to be of maximal dimension (we have not yet seen that all Borels have to have the same dimension). Note that in any representation of $ G\to\rm{GL}(W)$, $ B$ acts on the flag variety $ \mathcal{F}(W)$ of $ W$, and since $ B$ is a closed, solvable connected group, it fixes a flag in $ \mathcal{F}(W)$. However, by the fixed point theorem it fixes a flag; the problem here is we wouldn’t necessarily get a bijection between $ G/B$ and the orbit of the flag (choose the trivial representation).

On the other hand if we take the representation $ \phi:G\to\rm{GL}(V)$ with $ L\subseteq V$ having exactly stabiliser $ B$, then $ B$ acts on the flag variety $ \mathcal{F}(V/L)$, and so there is a flag starting with $ L$ and this gives us that the stabiliser of the flag would be exactly $ B$.

Since we chose $ B$ to have maximal dimension, it follows that the orbit of the flag has minimal dimension, hence closed, and hence complete being a closed subvariety of a complete variety. This gives us the identification of $ G/B$ with a complete variety. Moreover, this variety and the orbit of $ [L]$ are isomorphic because they both satisfy the universal property for quotients.

Example

From here, tons of cool structural results come out to help us understand the structure of algebraic groups. But for now, we should first work out an explicit example of all of the above stuff to get a concrete idea of the complete variety $ G/B$.

So we shall work out an example for $ G = \rm{GL}_2$. and $ B$ the subgroup of upper triangular matrices. Recall that in Highlights 8 we showed in a good amount of detail that $ B$ is actually a Borel subgroup. So let us realise $ G/B$ concretely in this case.

In this case $ k[G] = k[T_1,T_2,T_3,T_4,\rm{det}^{-1}]$, the multiplication being given by the Hopf algebra morphism $ k[G]\to k[G]\otimes_kk[G]$ defined on generators by:

$ T_1\mapsto T_1\otimes T_1 + T_2\otimes T_3$
$ T_2\mapsto T_1\otimes T_2 + T_2\otimes T_4$
$ T_3\mapsto T_3\otimes T_1 + T_4\otimes T_3$
$ T_4\mapsto T_3\otimes T_2 + T_4\otimes T_4$

The upper triangular matrices are then defined by the ideal $ I = (T_3)$. A right-translation invariant subspace containing $ T_3$ can be read off as $ W = \rm{span}_k(T_3,T_4)$. Then $ M = W\cap I$ is just $ M = \rm{span}_k(T_3)$, which is luckily for us already a line.

Let

$ x = \begin{pmatrix} a & b\\ c & d\end{pmatrix}\in \rm{GL}_2$

be arbitrary. A quick calculation shows that the representation $ \rm{GL_2}\to\rm{GL}(k^2)$ is given by

$ \begin{pmatrix} a & b\\ c & d\end{pmatrix}\in\rm{GL}_2\mapsto\begin{pmatrix} a & b\\ c & d\end{pmatrix}$

Hahahaha! Realising $ \rm{GL}_2$ as a matrix group already gave us the representation we wanted. Still, it’s nice to see the theory on a simple example. So we can easily see now that $ G/B\cong \mathbf{P}^1$! Actually, that $ G/B\cong \mathbf{P}^1$ is a bit of a special example, as this only happens when the Weyl group $ W\cong \mathbb{Z}/2$.

Leave a Reply

Your email address will not be published. Required fields are marked *