Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group.

I reference the papers Ti: Tits - Buildings of spherical type and finite BN pairs; So: Solomon - The Steinberg character of a finite group with BN pair; CLT: Curtis, Lehrer, and Tits - Spherical buildings and the character of the Steinberg representation.

There are two sorts of objects that seem to go under the name of spherical building of $G$ over $k$, namely, in the terminology of [CLT], the "combinatorial building" that is an abstract simplicial complex, and the building defined in [CLT, Section 2] that is obtained by actually gluing together spheres.

A short form of my question, that reveals the essential difficulty, and where (I think) there is no difference between the combinatorial and spherical buildings: is the building of $\operatorname{SL}_2$ (either sort) the same as its flag variety, or is it the flag variety together with an extra point?

The simplices of the combinatorial building are parabolic subgroups of $G$, with an apartment consisting of the parabolic subgroups containing a fixed maximal split torus of $G$. The apartments of the CLT spherical building are again parameterised by maximal split tori, and a point of the apartment corresponding to $S$ is, by definition, a ray in the real vector space $X_*(S) \otimes_{\mathbb Z} \mathbb R$, where $X_*(S)$ is the lattice of cocharacters of $S$. In order for the apartment to be a sphere, as Section 1 of [CLT] makes clear that it is meant to be, a half-line must be viewed as spanned by a *non-$0$* cocharacter. I'll comment on why this is important in a moment.

To each such ray $b$ one can associate a parabolic subgroup $P(b)$ of $G$ containing $S$, and Proposition 6.1 of CLT shows that this allows the spherical building to be viewed as a geometric realisation of the combinatorial building. *However*, since we insist that our rays are spanned by non-$0$ cocharacters, $P(b)$ can never equal $G$ (because $G$ is semisimple).

According to [So, Section 2], this seems to be fine; Solomon defines (in not quite this language) the vertices of the combinatorial building to be maximal parabolic subgroups (here clearly meaning maximal *proper* parabolic subgroups—else there would always be only one!), so that, for Solomon, the simplices in the spherical building are *proper* parabolic subgroups. On the other hand, [Ti, Theorem 5.2] says that the combinatorial building should have as simplices all parabolic subgroups; and maybe Tits implicitly *means* to say ‘proper’ here, but (setting aside whether such imprecision is likely from Tits) [Ti, Theorem 5.2(ii)] explicitly says that this notion of combinatorial building is the same as the one obtained from a BN-pair in [Ti, Theorem 3.2.6], and the definition there also allows *all* subgroups, not just proper subgroups, containing (a conjugate of) the $B$ of a BN-pair as simplices.

How does one reconcile these various definitions?