Let $A$ be a Henselian local ring and let $G$ be a split reductive $A$-group. I'm interested in some notion of a "building of parabolic subgroups" for the group scheme $G$.

In my specific cases of interest $A$ is nonreduced, something like the finite ring $\mathbb Z/p^n\mathbb Z$.

**Background**:
Using results from SGA 3 (if I understand correctly), the classification of parabolic subgroups is much the same over $A$ as over a field. For instance:

- every Borel subgroup contains a split torus,
- the Borel subgroups that contain a given split torus are a torsor under the Weyl group, and
- the parabolic subgroups that contain a Borel subgroup are in bijection with subsets of a basis of the root system.

We can therefore define a "(spherical) building of parabolic subgroups" $B(G)$ of $G$ in the usual way, where facets are (proper) parabolic subgroups and apartments correspond to split maximal tori, as the sets of parabolics that contain the torus.

So after waving our hands and using a major black box (SGA 3), it seems that we have defined some interesting building-like object in great generality. In fact, it seems like $B(G)$ just *is* a building on the nose: if we glance at the three building axioms (see e.g. Abramenko and Brown, Def. 4.1), $B(G)$ seems to satisfy them all.

**The bad news**: there is a classification of spherical buildings of rank $\geq3$: they arise from semisimple algebraic groups *over a field*.

So it seems very unlikely that $B(G)$ can be a building after all: it would have to magically agree with the building of some other group $G'$ over a field. ~~Looking at the case where $A$ is finite, I struggle to imagine how this could be~~. (See below.) But these buildings are also hard for me to get my hands on.

In fact, here is a proof, using the classification theorem, that $B(G)$ cannot be a spherical building in the case where (say) $A=\mathbb Z/p^n\mathbb Z$ and $\text{rank}(G)\geq3$. Let $X$ be the scheme of Borel subgroups, the maximal facets of $B(G)$. Using the Bruhat decomposition, and the fact that its strata are affine, one can see that

- $p\nmid|X(\mathbb F_p)|$ and
- the fibers of the reduction map $X(A)\to X(\mathbb F_p)$ have cardinality $p^{nD}$, so that
- $|X(A)| = p^{nD}\cdot|X(\mathbb F_p)|$,

where $D=\dim X_{\mathbb F_p}$. Hence the number of maximal chambers of $X$ is a multiple of a power of $p$, but this doesn't happen for the building of a reductive group over $\mathbb F_p$ (or $\mathbb F_q$).

**Questions**: (1) (Why) does $B(G)$ fail to satisfy the building axioms? (2) Has this simplicial complex $B(G)$ been studied before?

**Stray thought**: We have to be careful in working with group schemes or non-algebraically closed fields to differentiate between schemes and their sets of rational points. When $A=\mathbb F_2$ the definition I gave of an apartment is wrong if we interpret "contains a torus" on the level of rational points: a split torus over $\mathbb F_2$ does not have very many rational points.

Spherical buildings and the character of the Steinberg representation. They work over a field, though. I'm not sure if this building has been previously studied over a ring. (I'm making this a comment because it's not clear to me from your question whether you didn't know about this case, or knew about it and were wondering specifically about coefficients in a ring.) $\endgroup$