[Math] Why are parabolic subgroups called “parabolic subgroups”

gr.group-theoryho.history-overviewrt.representation-theoryterminology

Over the years, I have heard two different proposed answers to this question.

  1. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a really convincing explanation along these lines.

  2. "Parabolic" is short for "para-Borelic," meaning "containing a Borel subgroup."

Which answer, if either, is correct?

A related question is who first introduced the term and when. Chevalley perhaps?

Best Answer

It appears that neither of the answers is fully correct. There is a great book, "Essays in the history of Lie groups and algebraic groups" by Armand Borel, when it comes to references of this type. To quote from chapter VI section 2:

...There was no nice terminology for the subgroups $P _I$ with lie algebra the $\mathfrak p _I$ until R. Godement suggested calling them parabolic subgroups. I shall therefore anachronistically call them that...

"The geometry of the finite simple groups" by F. Buekenhout is on the other hand the only paper that came up in a search for paraborelic, and the author mentions he is using this term instead of parabolic to distinguish from parabolic subgroups of Chevalley groups.

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