Why is $\operatorname{SL}_2(q)/B\simeq\mathbb{P}^1(\mathbb{F}_q)$ as $\operatorname{SL}_2(q)$-sets?

Question

Let $G=\operatorname{SL}_2(q)$ be the special linear group over $\mathbb{F}_q$, and let $B$ be the standard Borel subgroup of upper triangular matrices. Why is $\operatorname{SL}_2(q)/B\simeq \mathbb{P}^1(\mathbb{F}_q)$ as $\operatorname{SL}_2(q)$-sets?

This identification shows up in showing that the closure of the Drinfeld curve in $\mathbb{P}^2(\overline{\mathbb{F}_q})$ is smooth, so I'm just curious as to what it is. It seems $B$ should act trivially on $\mathbb{P}^1(\mathbb{F}_q)$, but the obvious action $$ \begin{pmatrix} a & b \\ 0 & a^{-1}\end{pmatrix}\cdot[x:y]=[ax+by:a^{-1}y]=[a^2x+aby:y] $$ doesn't seem trivial.

Answer 1

If you want to be completely concrete, you can define a function $$ G/B\to\mathbb{P}^1(\mathbb{F}_q):\begin{pmatrix} a & b \\ c & d\end{pmatrix}B\mapsto [a:c]. $$

This is well-defined, since right multiplication by a matrix in $B$ scales the first column, but this is irrelevant in the image in the projective line. It is also a $G$-equivariant map, since the $G$-action is by matrix multiplication on both sets, and you essentially only care what happens to the first column.

It's easy to show this map is surjective, elements of the Borel subgroup map to $[1:0]$, $\begin{pmatrix} 0 & -1 \\ 1 & 1\end{pmatrix}$ maps to $[0:1]$, and matrices of the form $\begin{pmatrix} a & 0 \\ c & a^{-1}\end{pmatrix}$ hit everything else, for example. It's injective since both $G/B$ and $\mathbb{P}^1(\mathbb{F}_q)$ have cardinality $q+1$.

Answer 2

For any field $K$ $\text{GL}_2(K)$ and so $\text{SL}_2(K)$ act on the projective line $\Bbb P^1(K)$, considered as non-zero column vectors with scalar multiples identified. In both cases the action is transitive and the stabiliser of the class of the column vector $\pmatrix{1\\0}$ is the set of non-singular triangular matrices in GL${}_2$ and matrices $\pmatrix{a&b\\0&a^{-1}}$ in SL${}_2$.

I think you need to think of your $[x:y]$ as a column vector, not a row vector.