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.