TITLE: What is a (generalized) BN-pair?
QUESTION [11 upvotes]: Let us consider $GL_n(K)$ over a local field $K$. It has standard subgroups $N$ and $B$. $B$ is Iwahori subgroup, $N$ consists of monomial matrices. The pair comes close to a romantic ending, i.e. forming a BN-pair, as one gets a building of affine type out of it. However, it fails axiom (BN2) asking $sBs^{-1}\not = B$. You can also observe other things are out of order: maximal parabolics of different type are conjugate, etc...
What is the set of axioms governing this generalized BN-pair that alows all the usual building geometry to take place? Is there a name and other meaningful examples?
Please, help, I cannot sleep all day because it bothers me greatly.

REPLY [4 votes]: The notion of groups with a BN-pair has been generalized to groups with a root group datum. There is a wonderful paper by Pierre-Emmanuel Caprace and Bertrand Rémy on this topic, which you can download for free:

P.-E. Caprace & B. Rémy, "Groups with a root group datum", Innov. Incidence Geom. 9 (2009), 5-77.

It has a fairly long introduction (about 6 pages) which is really worth reading, containing a lot of information about the history of the different related concepts, and explaining how they are related with the theory of buildings, and so on.