# Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about.

Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' "Buildings of Spherical Type and Finite BN-pairs). Then $(B,N)$ has associated to it a chamber complex $X$ with a number of nice properties; most specifically, $X$ is connected, each apartment $\Sigma$ is a Coxeter complex, and $G$ acts transitively on the set $\{(\Sigma,\, C)\}$ of pairs where $\Sigma$ is an apartment and $C \in \Sigma$ is a chamber.

When $G$ is a semisimple group over a $p$-adic field $K$, there is a $BN$-pair where $B$ is an Iwahori subgroup.

More generally, when $G$ is reductive, there is no such $BN$-pair; there is, however, an generalized $BN$-pair.

My general question is:

Is there a connected building $X$ associated to a generalized $BN$-pair; that is, a connected chamber complex $X$ with an action of $G$ that is transitive on the set $\{(\Sigma,\, C)\}$ as above, and where each apartment $\Sigma$ is a Coxeter complex?

More specifically, if $G$ is a reductive group over a $p$-adic field, is there a canonical way to construct a chamber complex $X$ with the above properties, such that each Iwahori subgroup fixes a unique chamber?

For example, if $G = GL_2(\mathbb{Q}_p)$, then there is a generalized $BN$-pair where $B$ is the standard Iwahori subgroup, the Weyl group $W$ is generated by $w_1 = \pmatrix{0 & 1 \\ 1 & 0}$ and $w_2 = \pmatrix{0 & p \\ p^{-1} & 0}$, and $\Omega$ is generated by $s = \pmatrix{0 & p \\ 1 & 0}$ (here I am using the notation given at the beginning of Iwahori's Generalized Tits Systems on $p$-adic Semisimple Groups). One way I can think to build a building is as follows: $X$ is one-dimensional, where the set of vertices is $G/I_1 \cup G/I_2$, where $I_j$ is the parahoric generated by $Bw_jB$. The set of edges corresponds to $G/B$, and there is the obvious $G$-action. This, however, is disconnected since it is a countable disjoint union of trees.

• The 'enlarged building' associated to a reductive group is just the building of its derived group, times an affine space under the (real-ified) cocharacters of the centre. Commented Jan 12, 2015 at 21:36
• Also, the (enlarged) building of $\operatorname{GL}_2(\mathbb Q_p)$ is indeed 1-dimensional, but its set of vertices is $\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p)$. There is a very nice senior thesis by Joe Rabinoff at people.math.gatech.edu/~jrabinoff6/papers/building.pdf, describing all this in modern and very accessible language. Commented Jan 12, 2015 at 21:40
• @LSpice thanks for the comment; Rabinoff's notes are indeed excellent. Could you please expand on the graph structure of the enlarged building of $GL_2(\mathbb{Q}_p)$? I must be missing the identification between the set of vertices and $GL_2(\mathbb{Q}_p)/GL_2(\mathbb{Z}_p)$. Commented Jan 13, 2015 at 16:11
• A vertex $g\operatorname{GL}_2(\mathbb Z_p)$ should be identified with the maximal parahoric subgroup $g\operatorname{GL}_2(\mathbb Z_p)g^{-1}$. Two such parahorics are adjacent if their intersection is an Iwahori subgroup of each. (For example, $\operatorname{GL}_2(\mathbb Z_p)$ is adjacent to its conjugate by $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$.) Commented Jan 13, 2015 at 16:50

You may either work with an extended building as L. Spice suggests in his answer, or consider the following based on the fact that a generalized $BN$-pair contains a genuine $BN$-pair. With you notation, the group $N$ writes as a semidirect product $\Omega\ltimes N_0$ for some invariant subgroup $N_0$ of $N$. Then $G_0 = BN_0 B$ is a group with a true $BN$-pair $(B,N_0 )$, and we have $G = \Omega \ltimes G_0$. We may form the building $X$ for $G_0$ with respect to the $BN$-pair $(B,N_0 )$. Recall that $X$ is a simplicial complex whose vertices are in $G_0$-equivariant bijection with the parahoric subgroups $P$ of $G_0$. The action of $G$ by conjugation permutes the parahoric subgroups of $G_0$, whence induces a simplicial action of $G$ on $X$.