Geometric interpretation of $BN$-pairs

Question

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).

$[\ldots]$

Let $\mathcal{P} = \{P_1, \ldots, P_n\}$ be a minimal parabolic system for a group $G$, $B=P_1 \cap \ldots \cap P_n$ the Borel subgroup. A subgroup $N$ of $G$ is called a Weyl group for $\mathcal{P}$ iff

1) $N= \langle x_1, \ldots, x_n \rangle, x_i \in P_i-B, x_i^2 \in B$.

2) $B \cap N$ is a normal subgroup of $N$.

3) $N \cap P_i = (B \cap N) \langle x_i \rangle , i = 1, \ldots, n$.

$[\ldots]$

If additionally we have

(A) $G = BNB$ and

(B) $BgBhB \subset (BgB) \cup (BghB)$ for all $g,h \in N$

then we have a $BN$-pair. Geometrically, the Weyl group $N$ is the stabilizer of an apartment $\Delta$ of the geometry $\Gamma$ defined from $\mathcal{P}$, and $B$ is the stabilizer of a chamber of $\Delta$. I am trying to get a clear geometric view of these objects.

Here is my question. What could be a geometric interpretation of condition (B)?

Answer 1

Note: I'm using the general definition of BN-pair, which is weaker than the condition (B) given above.

It's a triangle inequality. One way to think about BN-pair is that they give a sort of combinatorial distance function on $G/B$. Given two cosets $g_1B$ and $g_2B$, you look at the product $Bg_1^{-1}g_2 B\in B\backslash G/B\cong N/(N\cap B):= W$ and think of that as the "distance" between them. To get a more numberish distance, you can let the length of an element of $W$ be the length of the shortest product of $x_i$'s which gives it.

If I take two cosets $BgB$ and $BhB$, and expand those as $Bx_{i_1}\cdots x_{i_n}B$ and $Bx_{j_1}\cdots x_{j_m}B$, then $$BgB\cdot BhB\subset Bx_{i_1}\cdots x_{i_n}B\cdot Bx_{j_1}\cdots x_{j_m}B\subset Bx_{i_1}B\cdots Bx_{i_n}Bx_{j_1}B\cdots Bx_{j_m}B.$$ Applying (B) inductively, we see that the last term is in the union of certain double cosets which have length shorter than the sum of that of $BgB$ and $BhB$.

To apply this to the "distance function," note that $Bg_1^{-1}g_3B\subset Bg_1^{-1}g_2B\cdot Bg_2^{-1}g_3B$, so the length of the distance between $g_1$ and $g_3$ is less than the sum of the lengths for $g_1$ to $g_2$ and $g_2$ to $g_3$: the triangle inequality.

Of course, that just shows that (B) (when $h=x_i$) implies the triangle inequality, but it's easy to see that it's also a special case: assume that $g'\in gBx_iB$. The correct form of the triangle inequality says that the distance from $B$ to $gB$ is obtained by deleting $x_j$'s from an expression for $g$, followed by $x_i$. On the other hand, $g\in g'Bx_iB$, so if I take an expression for the distance from $B$ to $g'B$, this is obtained from taking the expression for $g'$ followed by $x_i$ and deleting $x_j$'s. Some Coxeter group magic should tell you that either $BgB=Bg'B$ or $BgB=Bg'x_iB$, but I don't quite see it at the moment.

Answer 2

As a further commentary beyond what Ben has said, I'd emphasize that Stroth (along with Ronan and other finite group theorists) has modified some of the BN-pair and building formalism introduced by Tits. My understanding is that this is done partly in an attempt to unify the study of sporadic simple groups and simple groups of Lie type within the kind of "geometric" setting formulated first for the latter groups.

In any case, the broader notion of "parabolic system" in a group as used here is motivated by the earlier Lie structure but requires some experimentation with additional axioms beyond what Tits did. As Ben points out, the condition (B) you quote from that 1990 Durham conference article by Stroth goes beyond the conventional BN-pair axiom. In that conventional setting, which is close to the geometry of buildings and apartments, the Weyl group and its length function play a vital role in talking about distances in the geometry, etc. This is partly encoded in the usual version of condition (B).

Whether or not the structures studied by Stroth really add "geometric" flavor to the finite groups of interest is more than I can judge, but this does get outside the conventional framework of buildings with finite Weyl groups. By the way, it can do no harm to digest some of the original Tits thinking about the subject formulated as a detailed series of exercises for Section 2 of Chapter IV in the 1968 Bourbaki Chapters IV-VI of Groupes et algebres de Lie (later published in English translation by Springer). Naturally much of this shows up in his own Springer lecture notes on the "spherical" case as well as in later books on buildings.