TITLE: Buildings associated to generalized $BN$ pairs
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: 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 $X$.