Coxeter groups - Parabolic subgroups

Question

In the theory of Coxeter groups there is the notion of so called parabolic subgroups. I'm wondering is this term just a random name, or there are some historical reasons? Why parabolic? Thanks.

Answer 1

The history is definitely somewhat convoluted. Note first that the term "Coxeter group" itself was introduced by Bourbaki in their 1968 volume containing chapters 4-6 of Groupes et algebres de Lie. The first section of Chapter 4 studies Coxeter systems $(W,S)$ (with $S$ typically finite) in great generality, inspired in part by Coxeter's work on reflection groups but especially by the connection made by Witt and others with Lie theory. Finite Coxeter groups mostly arise as "Weyl groups" in classical semisimple Lie theory, while affine Coxeter groups (= affine Weyl groups) come up less directly in various areas of Lie theory.

In the general study of Coxeter groups, one discovers that the subgroup $W_X$ generated by a subset $X \subset S$ is in a natural way a Coxeter group. Bourbaki goes on in the second section of Chapter 4 to study $BN$-pairs (or "Tits systems"), which capture some of the essential structure of semisimple Lie groups or algebraic groups or finite groups of Lie type. Here the subgroups $W_X$ are naturally attached to what Bourbaki calls (in section 2.6) "parabolic" subgroups" of a group $G$ with a Tits system and "Weyl group" $W$ (a Coxeter group).

The use of the term "parabolic" in semisimple Lie groups or linear algebraic groups is itself inspired quite indirectly by the older study of the modular group in classical function theory. (This at any rate is the kind of answer given to the terminology question raised with Mostow at a conference I attended decades ago.) The definition of "parabolic subgroup" in Lie theory is just a matter of convenience, I think: it is a closed subgroup containing some Borel subgroup (another term of convenience, to avoid clumsy expressions like "maximal closed connected solvable subgroup").

By now the use of the odd word "parabolic" for a subgroup $W_X$ of a Coxeter group $W$ is purely conventional but widespread.