The projective line as a rank-one flag variety

Statement

Let $k$ be a field and let $B$ be the split upper-triangular Borel subgroup over $k$ . The orbit map that sends a matrix to the line spanned by its first column induces

SL_2/B_{SL_2}\;\simeq\;\mathbb P^1_k \qquad\text{and}\qquad PGL_2/B_{PGL_2}\;\simeq\;\mathbb P^1_k.

Thus the projective line is the complete flag variety in type $A_1$ .

What this does and does not identify

The displayed isomorphisms concern homogeneous spaces. They do not identify $\mathbb P^1$ with either group. Over $\mathbb C$ , $PGL_2(\mathbb C)$ also acts as the holomorphic automorphism group of the Riemann sphere, while in Langlands duality $PGL_2$ occurs as the dual group of $SL_2$ . These are different roles of the same algebraic group.

References

T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, 1998, Chapter 8. DOI.