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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?

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I am certain that it's #1, but the terms "parahoric" (containing an Iwahori subgroup) and "mirabolic" (miracle parabolic) were so named to be consonant with "parabolic", which may have led to the folk etymology you've described. – Victor Protsak May 17 '10 at 3:38
Wow, the second one would be very creative. – user717 May 17 '10 at 9:00
The invention of "parahoric" (after Iwahori) is apparently due to Bruhat-Tits in their follow-up work on structure theory over local fields following fundamental work by Iwahori and Matsumoto. Tits has always been fond of this kind of wordplay. (The introduction of "Borel subgroup" in his 1965 paper with Borel was probably due to Tits, though they left that ambiguous in a famous footnote.) – Jim Humphreys May 17 '10 at 11:31
@Victor: What makes you so certain that it's #1? Any concrete evidence? I can imagine either definition being the original one and the other one being the folk etymology that was invented because it seemed plausible. – Timothy Chow May 17 '10 at 14:13
@JimHumphreys, I hadn't noticed the "famous footnote" before, but I guess you refer to the charmingly coy "L'un des auteurs insistant pour que l'on adopte cette terminologie, aujourd'hui généralement admise, l'autre auteur s'y résigne." on p. 669? – L Spice Jun 24 '15 at 14:26
up vote 15 down vote accepted

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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Borel's attribution of the terminology "parabolic subgroup" to Godement is reasonable, but Timothy Chow's first option probably comes closest to the rationale behind this choice. Study of the modular group by Fricke, Klein, and others distinguished several types of elements: "elliptic", "hyperbolic", "parabolic" (the latter typically coming from unipotent matrices). When Dan Mostow was asked about the origin of the naming convention back in 1977, I recall that he attributed it to the parallel with modular groups and parabolic elements. By 1962 Tits was using the term in his papers. – Jim Humphreys May 17 '10 at 11:25
P.S. A late instance of "parabolic" in connection with the modular group occurs in a 1974 thesis at NYU by the last student there of Wilhelm Magnus: Nonparabolic Subgroups of the Modular Group by Carol Tretkoff. But in line with Benoit's answer, the underlying rationale for the usage comes from study of homogeneous spaces such as $G/P$ in Lie theory. Borel himself didn't use the term "parabolic subgroup" in his 1956 Annals paper, but focused on complete/projective varieties starting with $G/B$. By 1962 he as well as Tits and others were using the term in print. – Jim Humphreys May 17 '10 at 13:03

My (completely non historical) point of view is the following. When you study non-compact symmetric spaces, e.g. the real hyperbolic space, isometries can be divided into three classes: elliptic (fixing a point in the space, so that it generates a relatively compact subgroup), hyperbolic (translates a geodesic, and acts like a dilation on the boudary of the space), and parabolic (none of the preceding type, but can be approximated both by elliptic and hyperbolic elements; always fixes a point on the boundary). In this context, a parabolic subgroup is the stabilizer of a point of the boundary, and contains many parabolic elements.

I guess that in a more algebraic (or should I say less geometrical?) context, this notion might generalize naturally to what is actually called a parabolic subgroup.

I hope this at least clarifies what is often meant by your answer #1.

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This is roughly what I have heard before, but the reason it hasn't struck me as being a clincher is that the connection you give between parabolic subgroups and parabolic elements is not as crisp as I would have expected if this were the true motivation for the terminology. Is there a sharper theorem here than "a parabolic subgroup contains many parabolic elements"? – Timothy Chow May 17 '10 at 14:18
@Timothy: You may be expecting more rationality in the choice of terminology than exists. It's usually hard to come up with just the right word (standard or invented), so people may rely on (1) bland choices like "normal", (2) words transplanted from their original context like "parabolic", (3) names of people (appropriate or not) somehow associated with the concept --- the invented term "K3 surface" is one variant. After a while it's too late to go back and rethink the choices, as Freudentahl-deVries tried to do using highly nonstandard terminology in their 1969 book Linear Lie Groups. – Jim Humphreys May 17 '10 at 17:26
@Timothy Chow: in the stabilizer of a point $p$ of the boundary, there are: -- all hyperbolic elements whose translated geodesic has $p$ as an endpoint; there are many ways to deform the geodesic so that the other endpoint also tends to $p$, and the resulted isometries in the limit are parabolic, -- all elliptic elements whose fixed point set contains $p$ in its closure; for example in the real hyperbolic case, such a fixed point set is a totally geodesic subspace, that can be deformed to $\{p\}$, the deformed isometries in the limit being parabolic. – Benoît Kloeckner May 17 '10 at 18:32
Another point, in the real hyperbolic space: the stabilizer of a boundary point is isomorphic to the set of similarities of the euclidean space of one less dimension. Inside this set, the translations are the parabolic elements. This makes many of them. More significantly, you form a cusp by quotienting the space by a lattice of this euclidean space: parabolic elements play in this respect a prominent rôle. – Benoît Kloeckner May 17 '10 at 18:33
@Jim: Wow, the table of contents alone in "Linear Lie Groups" is a barrel of laughs. With crazy-sounding terms like trunks, tools ("Weyl tool"), dressings, wrappings, and virtual reality (really), I can't fathom what those guys were smoking when they wrote the book. – BCnrd May 18 '10 at 6:43

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