I was wondering the rationale behind naming parabolic subgroups of linear algebraic groups. The answer, interestingly, comes from the action of $SL(2,\mathbb R)$ on the upper half plane. (I came up with this little discovery on my own.)

The orbit of the point $i$ under the action of the standard parabolic subgroup of $SL(2,\mathbb R)$ is a parabola.

The upper half-plane is an object that comes up in many parts of mathematics, hyperbolic geometry, complex analysis and number theory to name a few. The group $SL(2,\mathbb R)$ acts on it by fractional linear transformations:

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} .\ z \mapsto \displaystyle \left( \frac{az+b}{cz+d} \right) .$

A parabolic subgroup is it’s subgroup $P$ such that quotienting by $P$ gives a compact variety. Upto conjugation, the only parabolic subgroup of $SL(2,\mathbb R)$ is $\begin{pmatrix} * & * \\ 0 & * \end{pmatrix}.$ It’s action on the point $i$ is given by:

$\begin{pmatrix} x & y \\ 0 & x^{-1} \end{pmatrix} . i = \displaystyle \left( \frac{xi+y}{0i+x^{-1}} \right) = y + i x^2$

whose locus is a parabola.

I wonder why textbooks in algebraic groups don’t mention this!

EDIT: (21 April, 2015) The above calculation is WRONG. I don’t know the answer to “parabolic”.