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.

• I'm not sure what your starting point is, but the original treatment of BN-pairs in "Chevalley groups" over local fields (which more or less covers the general linear case) was done by Iwahori and Matsumoto in their 1965 IHES paper: numdam.org/numdam-bin/fitem?id=PMIHES_1965__25__5_0 Jun 27 '12 at 11:22
• P.S. Your formulation of axiom (BN2) needs the symbol "not equal" in place of "not a member of". Jun 27 '12 at 11:25
• Thanks, Jim, corrected. In a "Chevalley group" SL you have BN-pair but you cannot extend it GL... unless I have gone completely looney... Jun 27 '12 at 11:51

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.

• Bull-eye!!!!!!! Jul 23 '12 at 16:22

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.

• Actually, a group with a root group datum always also has a (twin) BN-pair (Theorem 4.1 in loc.cit). So I personally would rather say that group with BN-pairs generalize groups with root group datum... ? Jun 27 '12 at 14:31
• @Max: Good point; that makes my answer less relevant. (The contents of the paper might still be relevant, though.) Jun 27 '12 at 15:16
• I gather it is some kind of generalization of a Kac-Moody group, and it does have a proper BN-pair. It does not seem to help with $GL$ at slightest... Jun 27 '12 at 15:27
• Actually, the paper by Caprace still might be relevant to you. Say your local field is $\mathbb{F}_q((t))$ (Laurent polynomials over a finite field). Then this one of two possible completions of the ring of Laurent polynomials $\mathbb{F}_q[t,t^{-1}]$. Now, it turns out that the group $G^+ :=SL_n( \mathbb{F}_q((t)) )$ is a (topological) completion of the affine Kac-Moody group $G:=SL_n(\mathbb{F}_q[t,t^{-1}])$. Moreover, $G$ in general embeds as a lattice into $G^+\times G^-$, where $G^- :=SL_n( \mathbb{F}_q((t^{-1})) )$. Jun 27 '12 at 15:42
• Yes, you can but then this bigger torus permutes vertices of different type in the building, while the torus in $SL$ preserves the type. Hence, my question... Jun 27 '12 at 15:52

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.

• Note that you will have to change the set of generators for the affine Weyl group a little bit while changing form $GL(2)$ to $SL(2)$. Jun 27 '12 at 13:10
• Note that the center of a group with BN-pair always acts trivially on the associated building. So in a sense, the building theoretic part (and hence the BN-pair) only sees'' the original group modulo its center. So this "cheap trick" indeed seems quite sensible, I guess... Well, depending on what exactly you want to do ;). Jun 27 '12 at 14:35
• I am not sure what your magic $Z(F)$ is, doc, but I am quite sure that $Z(GL)\cdot SL$ is strictly smaller than $GL$. The determinant of a matrix in the former is always an $n$-th power!! In fact, the (Iwahori,Monomonial) define a BN-pair on this group, but not on the whole $GL$ Jun 27 '12 at 15:23
• @Bugs Bunny: This was certainly a mistake. You precisely have to change the set of generators for the Coxeter group because of that fact. Jun 28 '12 at 7:25