spherical buildings for non-split groups

Question

I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of split groups in many sources (e.g. in Garrett's book), but there is hardly anything available in the non-split case. I know how to do so in the pseudo-orthogonal case (groups O(p,q)). In principle, I could run similar arguments for other types (unitary groups over complex numbers, quaternions, etc.), but it is a bit tedious, so I am looking for a reference.

Answer 1

Misha, Tits' Lecture Note "Buildings of spherical type and finite BN pairs" gives a fairly explicit description of the buildings associated to the classical groups (not just the split ones). I also wrote a survey called "Buildings and classical groups" (arXiv:math/0307117) which appeared in print some years ago. These sources deal with general fields and division rings. Of course, some details become easier over the reals, the complex numbers and the quaternions, partly because one does not have to worry about characteristic 2.

The buildings associated to hermitian or symplectic forms (and hence to orthogonal, unitary or symplectic groups) are also called polar spaces. Under this name, one can find lots of references.

Here are some more details, based on Section 2.7 in my article arXiv:math/0307117. Since we are interested in Lie groups, we are dealing with finite dimensional modules over the reals, the complex numbers or the real quaternions.

We are given a nondegenerate $(\sigma,\epsilon)$-hermitian form $h$ on a finite dimensional right module $V$ over $\mathbb K=\mathbb R,\mathbb C,\mathbb H$. This means that $\epsilon=\pm 1$, that $\sigma$ is an involution (an anti-automorphism whose square is the identity) on $\mathbb K$ and that $h$ is a $\sigma$-sesqulinear form on $V$, with $$h(u,va)=h(u,v)a=\epsilon h(va,u)^\sigma\qquad\forall u,v\in V,\ a\in\mathbb K$$ and with $V^\perp=0$. A submodule $W$ is called totally isotropic if $W\subseteq W^\perp$. The building is the simplicial complex consisting of all (partial) flags consisting of totally isotropic subspaces. If we rescale the hermitian form $h$ by multiplying it by a (suitable) nonzero scalar, the building is obviously not changed. Up to this rescaling process, there are the following cases.

1) Symplectic groups over $\mathbb R,\mathbb C$, where $\sigma=id_{\mathbb K}$ and $\epsilon=-1$.

2) Orthogonal groups over $\mathbb R,\mathbb C$, where $\sigma=id_{\mathbb K}$ and $\epsilon=1$.

3) Unitary groups over $\mathbb C,\mathbb H$, where $a^\sigma=\bar a$ is the standard conjugation and $\epsilon=1$.

4) Unitary groups over $\mathbb H$, where $a^\sigma=-i\bar a i $ and $\epsilon=1$.

In case 1) and 2), the building has simplicial dimension $m-1$, were $\dim(V)=2m$.

In case 3) the building has simplicial dimension $m-1$, and the form $h$ has signature $(m,m+k)$ for $k\geq 0$.

In case 4), the building has simplicial dimension $m-1$, and $\dim V=2m$ or $\dim V=2m+1$.

In case 2), if $k=0$, then the building is not thick. If one want a thick building, one has to consider instead the "oriflamme geometry" which is explained for example in Tits' Lecture Note.