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”.

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