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

The orbit of the point under the action of the standard parabolic subgroup of 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 acts on it by fractional linear transformations:

A parabolic subgroup is it’s subgroup such that quotienting by gives a compact variety. Upto conjugation, the only parabolic subgroup of is It’s action on the point is given by:

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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April 30, 2015 at 01:36

chandrasekharDear Abhishek,

Looks like the calculation is wrong. Shouldn’t it be xy + ix^2?? You have an x inverse in the denominator so numerator gets multiplied by x.

April 30, 2015 at 10:04

Abhishek ParabYou’re right, look at my comment at the end of the post. I didn’t delete the post because I’m hoping someone can tell me the right answer.