6
$\begingroup$

A subgroup $B < G$ of a connected algebraic group $G$ that is maximal among the solvable connected subgroups is called a Borel subgroup. A closed subgroup $P < G$ is parabolic if it contains a Borel subgroup.

Let $K/k$ be a field extension with $K$ algebraically closed. Here they say that minimal parabolic $k$-subgroups (i.e. defined over $k$) play the same role over $k$ as the Borel subgroups play over $K$.

How should I think of this? Are there any examples of groups, where the minimal parabolic $k$-subgroups are not Borel subgroups? I guess this would mean that the Borel subgroups must not be defined over $k$?

I am mostly interested in $K= \mathbb{C}, k = \mathbb{R}$.

$\endgroup$

1 Answer 1

4
$\begingroup$

Unfortunately, the linked anonymous Encyclopedia of Mathematics article does a terrible job at explaining this. It is fine if one works over algebraically closed fields (when Borel is the same as minimal parabolic) but not otherwise. My suggestion is to read two sources.

The first one is:

Humphreys, James E., Linear algebraic groups. Corr. 2nd printing, Graduate Texts in Mathematics, 21. New York - Heidelberg - Berlin: Springer-Verlag. XVI, 253 p. DM 72.00; $ 34.30 (1981). ZBL0471.20029.

It is still suboptimal since Humphreys does not always say explicitly if his field is algebraically closed or not. But Chapters 34.5 and, especially, 34.6 clarify things, especially his example in Chapter 34.5 where he works out the example of an orthogonal group over a field which is not algebraically closed. In this example you get a minimal parabolic which is not Borel. In particular, this example shows that you simply cannot define $k$-parabolic subgroups as those containing $k$-Borel!

As a differential geometer, my favorite source is

Eberlein, Patrick B., Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics. Chicago, IL: The University of Chicago Press. 449 p. (1996). ZBL0883.53003.

In the book he works with real Lie groups $G$ (just as you asked) and defines parabolic subgroups geometrically as $G$-stabilizers of simplices in the (spherical) Tits building of the symmetric space $G/K$ (the spherical building is defined via geodesics in $X$). This is the interpretation given in the penultimate paragraph of the linked Encyclopedia of Mathematics article.

With this in mind, here is a geometric interpretation of Humphreys example, in the special case when $r=1$. Let $X$ be the hyperbolic $n$-space and $G$ the group of isometries of $X$, i.e. $PO(n,1)$. Assume that $n\ge 4$. Then each parabolic subgroup $P< G$ is the stabilizer of a point of the ideal boundary sphere of $X$. Since rank is 1, these are also minimal parabolic subgroups. Then, $P$ is the semidirect product of a solvable subgroup and the orthogonal group $O(n-1)$. Thus, $P$ is not solvable since $n\ge 4$. The solvable group in turn is the semidirect product of ${\mathbb R}^{n-1}$ (a maximal connected unipotent in $G$) and ${\mathbb R}_+$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .