What is a (generalized) BN-pair? 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.
 A: The axiom system you want for a "generalized BN-pair" has been written down concisely by Iwahori in the proceedings of the 1965 AMS Summer Institute
titled Generalized Tits systems (Bruhat decomposition) on p-adic semisimple groups.  At the end of Section 2 he makes the transition to general linear groups explicit.
The whole subject of BN-pairs and buildings started with Tits, but the emphasis on local fields began with the work of Iwahori and Matsumoto in the mid-1960s.   This was followed by an extensive development for reductive groups over various fields of definition (Bruhat-Tits, Prasad, and others).    But already in the early days there was a conscious concern about the transition from narrowly defined "Chevalley groups" (originally of adjoint type) and simply connected semisimple groups to general linear groups and others which are not simply connected.
Anyway, it's a large literature by now, including books on buildings, which somewhat obscures these special cases of interest.
ADDED: Besides the two short articles by Bruhat and Iwahori in the AMS volume 9 of Proc. Symp. Pure Math. from the 1965 Boulder institute, there is a longer survey by Tits from the 1977 Corvallis summer institute, published in Part I of volume 33: Reductive groups over local fields, pages 29-69.    (Both of these old conference proceedings used to be available online at the AMS website but have since apparently disappeared.)     As noted above, later literature expands the framework considerably and deals more explicitly with buildings.   But these early surveys are still helpful.
A: 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.
A: Usually one talks about the generalized BN-pair, when you have non-compact center. I am not sure exactly what the definitions are, but here is a cheap trick for $GL(2)$.
Enlarge the Iwahori subgroup by the center
$$I = B \cdot Z(F),$$
then you get a nice BN pair.
We have to change one generator $\begin{pmatrix} 0 & \pi \newline \pi^{-1} & 0 \end{pmatrix}$ to $\begin{pmatrix} 0 & \pi \newline 1 & 0 \end{pmatrix}$, precisely for the matter you mention in the comments. Note that $\begin{pmatrix} 0 & \pi \newline 1 & 0 \end{pmatrix}^2 \in Z(F)$ of $GL(2)$.
I have not played with $n >2$ so far. But probably for $GL(n)$, something similar is possible.
Large center is pretty annoying sometimes, but does not really introduce new phenomena.
Another cheap trick is to work out the theory for
$$ GL^1(n,F) = \{ g : | \det g|=1 \},$$
which has compact center, but I prefer the pasting the center approach. 
Of course you can also look at $PGL(n)$, but this feels slightly different than caring the center along. The building should give you some control about the representation theory of the group in question, and there are certainly smooth admissible representation of $GL(n,F)$, where no twist has trivial central character.
