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.



What is a (generalized) BN-pair?
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.


