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Let $\mathbb{K}$ be an arbitrary field, and $G$ an algebraic group (or group variety?) over this field. A Borel subgroup of $G$ is a connected solvable subgroup variety $B$ of $G$ such that $G/B$ is complete. A parabolic subgroup of $G$ is a subgroup variety $P$ such that $G/P$ is complete.

From section 29.4 of Tauvel and Yu, I know that if $\mathbb{K}$ is algebraically closed and has characteristic zero, then a Borel subalgebra of a finite-dimensional Lie $\mathbb{K}$-algebra $\mathfrak{g}$ (not necessarily semisimple) is defined to be a maximal solvable subalgebra $\mathfrak{b} \leq \mathfrak{g}$. A parabolic subalgebra $\mathfrak{p} \leq \mathfrak{g}$ is any subalgebra containing a Borel subalgebra.

Tauvel and Yu prove that under these hypotheses on the field ($\mathbb{K}$ is algebraically closed and has characteristic zero), if $G$ is a connected algebraic group with Lie algebra $\mathfrak{g}$, then parabolic subalgebras of $\mathfrak{g}$ are precisely the Lie algebras of parabolic subgroups of $G$.

Now if we generalize so that $\mathbb{K}$ is not algebraically closed (or more generally, but less importantly, when the characteristic is nonzero), I would like to know to what extent these definitions and correspondences carry over? Moreover, if $\overline{\mathbb{K}}$ is the algebraic closure of $\mathbb{K}$, what properties are preserved and reflected by "extending" the scalars?

For my particular purposes, I am most interested in the case $\mathbb{K}=\mathbb{R}$. However, it seems that everywhere I look the complex case is emphasized. I like to think very categorically, and in a "top-down" sort of way, so I am hoping to get a good overview of where I am going and what to expect before I slog through the complex theory.

If it helps at all, I trying to get a handle on parabolic subgroups and subalgebras so that I can reach a more comprehensive understanding of parabolic Cartan geometries and (eventually) BGG resolutions.

Unfortunately for me, my knowledge of algebraic geometry and algebraic groups is very limited, so I apologize if the answer to my question is very obvious.

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    $\begingroup$ Indeed, the general theory is described best (imho) in the language of algebraic groups, so at some point you might want read a text dealing with these. However, maybe the first thing to do would be to work through a single illustrative example which you understand. I would start with $\text{SO}(3,1)$. $\endgroup$ – Uri Bader Aug 3 '16 at 11:27
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    $\begingroup$ My recommendation would be proceed in two paralile ways (that might meet before ininity). 1. Read a text on algebraic groups. For this it might be actually a good idea to start by the algebraically closed case, and not jump ahead. 2. To read a text in Lie theory that compares the real and complex cases. I am not sure which right now. Maybe Knapp, Varadarjan, Helgason... $\endgroup$ – Uri Bader Aug 3 '16 at 12:00
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    $\begingroup$ "or more generally, but less importantly, when the characteristic is nonzero" For your purposes this may be less important, but in general far more can fail in positive characteristic than in the non-algebraically closed case. For example, the Lie algebra of $\mathrm{SL}_2$ (reductive) in characteristic 2 is soluble. (It is a pretty safe bet that, if checking whether a property of Lie groups can be deduced from an analogous property of Lie algebras in positive characteristic, then the answer is probably 'no'. The proper definition of 'Borel subalgebra' here is 'algebra of Borel'.) $\endgroup$ – LSpice Aug 3 '16 at 13:39
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    $\begingroup$ To address your actual question, though, one major possible point of failure is that there need not be any Borel subgroup, or Borel subalgebra, over a non-algebraically closed field. $\endgroup$ – LSpice Aug 3 '16 at 13:40
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    $\begingroup$ Keep in mind that over an arbitrary field (even of characteristic 0), there are foundational questions about the meaning of "projective" and "complete" variety. As L. Spice points out, fields of definition also come up in questions about Borel or more general parabolic subgroups. Though the notion of parabolic subgroup has a history in Lie group theory, it became most prominent in work on algebraic groups by Borel and Tits in the 1960s. But here the language of algebraic geometry needs extra care. $\endgroup$ – Jim Humphreys Aug 3 '16 at 16:08
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Let me work through an example for you. Take a $(2,8)$ form on $V=\mathbb{R}^{10}$. Let $G=\text{O}(2,8)$ be the group fixing this form. You can find in $V$ isotropic lines, isotropic planes (but not higher dimensional isotropic spaces) and isotropic flags (a pair, isotropic line inside an isotropic plane). The parabolics in $G$ will be the stabilizers of such objects (and the stabilizer of 0, which is $G$ itself). In particular, you will have four types of parabolics. The stabilizer of a flag is minimal: every other parabolic contains such. You can write one explicitly and observe that it contains a normal solvable subgroup, and modulo it you get a compact group $(\simeq \text{O}(6)$, the isometry group of the perp of the isotroic plane mod its radical).

If I was to consider a $(5,5)$ form I would get $2^5$ types of parabolics, each being the stabilizer of some isotropic flag. This is the "split case". The parabolic pattern here is the same as in its complexification.

Note that upon complexifying the two cases above are conjugated: a conjugating matrix could be taken to be a diagonal matrix with 7 1's ant 3 $i$'s (this is wrt some natural choice of coordinates). You now see that some, but not all, of the parabolic types in the complexification are actually represented in $G$. These are "defined over the reals".

This generalizes. In the complexification you always have $2^R$ types of parabolics, $R$ is called the (split or complex) rank of $G$, each contains a minimal one and the minimal ones, called "Borel subgroups", are all solvable. In fact, a subgroup of the complex group is parabolic iff it contains a Borel subgroup. Among all parabolic types in the complexified group, only $2^r$ types are represented in $G$, $r$ is called the real rank of $G$. These are said to be defined over the reals. Each such parabolic contains a minimal such one, the minimal ones are always solvable by compact. In fact, a subgroup of $G$ is parabolic iff it contains a minimal parabolic.

An extreme case is when $G$ is compact. Then the only prabolic which is defined over the reals is $G$ itself. In this case the real rank of $G$ is 0. The other extreme, where $r=R$, is called "the split case".

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I think the best book for you is Parabolic Geometries I : Background and General Theory by Andreas Čap and Jan Slovák. The first half covers the basics of Cartan geometries and classification of real Lie groups. The parabolic Cartan geometries are the content of its second half. As far as I know, there is no book for BGG resolutions as of yet. My feeling is that we will have to wait a few years for new results "to stabilize". E.g. for the most advanced stuff in this direction you can checkout arXiv:1510.03331 and its sequel.

Some answers to your questions can be found in the third chapter of the aforementioned book. The main point here is that from the point of view of parabolic geometries you need to care only about subgroups of $G$ that preserve certain filtrations.

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