The Analog of Borel Subgroup in a Compact Real Form I recently learned that there is a one-to-one correspondence between isomorphism classes of complex reductive groups and isomorphism classes of compact connected real Lie groups given by taking a compact real form in one direction and complexification in the other direction. I am happy with many parallels such as the existence of maximal tori, the definition of a Weyl groups, the classification of finite dimensional irreducible representations by dominant integral weights and so on. (Please correct me if I get any of the statements above wrong since I am just pulling them out from my memory.)
One question that bothers me is the following. In the complex reductive case, given a fixed maximal torus one can choose a Borel subgroup containing that torus, which is equivalent to choosing a collection of simple roots in the root system; is there an analog of this in the compact connected real Lie group case? (I don't know how to properly define a root system for a compact connected real Lie group without passing to the corresponding complex reductive group either.) Thanks a lot.
 A: As the comments indicate, you have to go over to the complexified Lie algebra to study the simple roots and corresponding Borel subalgebras.   This is a traditional pathway, so I'll add (in community-wiki format) a couple of explicit references.    One was suggested already: the translated text by Brocker and tom Dieck, which covers both the structure theory and the (finite dimensional!) representation theory of a compact Lie group $G$ here.  
Early in the book they work directly with $G$, showing for example that each element lies in a conjugate of a fixed maximal torus $T$.   (In algebraic group language, every element of $G$ is "semisimple".)   But eventually the root system has to be studied using the complexified Lie algebra, with results brought back to the compact Lie groups.
For a concise but more formal treatment in a similar spirit, you might
want to compare Chapter 9 in Bourbaki's treatise Lie Groups and Lie Algebras (original French edition, 1982).
As Venkataramana observes, the flag manifold in the compact setting is a homogeneous space $G/T$, but in the complex Lie group setting it is realized as $G_\mathbb{C}/B$ for a Borel subgroup $B$.    Either way, the construction of finite dimensional irreducible representations is often carried out in terms of global sections of line bundles on this manifold in the spirit of Borel-Weil, relative to weights in a dominant Weyl chamber.            
A: It sounds like you want a datum to associate to a compact Lie group and chosen torus, that's functorially the same as a choice of Borel containing that torus if one were to complexify, without using the word "complexify" anywhere. 
If that's right, I suggest "choice of connected component of $(\mathfrak t^*$ with all points removed that have nontrivial $W$-stabilizer)".
