# Tits building of a linear algebraic group

I have a basic (probably naive) question about Tits buildings. Let $G$ be a (connected) linear algebraic group over a field $k$ (I am interested in the case where $k$ is algebraically closed but I appreciate information for general $k$ also).

When $G$ is a semisimple or reductive group, the (spherical Tits) building associated to $G$ is defined as the simplicial complex whose simplices correspond to parabolic subgroups of $G$ and apartments correspond to maximal tori. My question is, does this definition work for a general (connected) linear algebraic group as well? That is, does the collection of parabolic subgroups of $G$ form a building? If not, what is the main axiom that fails?

Also, say when $G$ is reductive, is it correct to think of the apartment corresponding to a maximal torus $T$ as the cocharacter lattice of $T$ (or lattice of (algebraic) 1-parameter subgroups)? In other words, can one think of the total space of a building as the set of all (algebraic) 1-parameter subgroups in $G$?

## 2 Answers

Yes, the definition you make works for a general linear algebraic group $G$. The reason you haven't seen it mentioned is that the solvable radical $S$ of $G$ is contained in any of its parabolic subgroups, thus the building associated with $G$ coincides with the building associated its natural semisimple factor $G/S$.

Your last paragraph is incorrect. Given a maximal torus $T$, the apartment associated with $T$ is the (finite) collection of all parabolic subgroups containing $T$ (while the set of cocharacters of $T$ is infinite) .

• Thanks @UriBader for the answer. For the last part, I do not mean that the 1-parameter subgroups are the vertices or simplices in the simplicial complex. What I have in mind is that (in the semisimple or reductive case) each apartment is a Coxeter complex and the chambers in a Coxeter complex can be regarded as Weyl chmabers (living in the cocharacter lattice of a maximal torus).
– Kiu
May 6, 2018 at 1:11
• So one can visualize the building as the collection of all 1-parameter subgroups of G and they are grouped into chambers (i.e. simplicial complex structure) according to which Weyl chamber of which maximal torus they lie in. I hope this explanation clarifies my point.
– Kiu
May 6, 2018 at 1:12
• @Kiu I undid the edit you made (namely deleting my second paragraph). This is my answer. You may not accept it if you don't like it. But an answer to a question containing the sentence "can one think of the total space of a building as the set of all (algebraic) 1-parameter subgroups in G?" will not be complete without pointing at the fact that this sentence, as stated, is wrong. May 6, 2018 at 8:52
• OK @UriBader, thanks anyways. I guess the answer to my last question is more or less trivially yes. It seems like people in the area of "buildings" (of algebraic groups) mainly care about it as an abstract simplicial complex. For applications (of the notion of building) I have in mind, it makes sense to visualize a chambers as "1-parameter subgroups that lie in the same Weyl chamber".
– Kiu
May 6, 2018 at 15:11
• No offense was intended. Sorry for any confusion. I accepted you answer.
– Kiu
May 7, 2018 at 14:51

I asked a related question Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group? recently, at which point MO reminded me of yours. @UriBader has already given a very nice answer to both questions, with the answer to the first question being surely all that could be wished; but perhaps it is helpful to note a sense in which the answer to your second question, while literally "no", is nonetheless almost "yes".

Namely, there is an object defined in CLT: Curtis, Lehrer, and Tits - Spherical buildings and the character of the Steinberg representation, called the spherical building of a reductive group $$G$$ over $$k$$, that, for $$G$$ semisimple, gives a geometric realisation of the sort of building you discuss (there called the combinatorial building). The CLT spherical building is made up of apartments indexed by the maximal split tori in $$G$$; and the points of the apartment corresponding to $$S$$ are, not literally cocharacters of $$S$$, but rays in the real-ised vector space $$X_*(S) \otimes_{\mathbb Z} \mathbb R$$, where $$X_*(S)$$ is the cocharacter lattice of $$S$$. See Section 1 for the definition of the apartments, Section 2 for the definition of the building, and Proposition 6.1 for the relationship to the combinatorial building. The proof of Proposition 6.1 is sketched in by reference to "the alternative definition of $$P(b)$$ in Section 1", which is essentially the viewpoint you proposed in your comment.

There is also a discussion in Section 7 about what happens in the general reductive case; the CLT spherical building and the combinatorial building treat the centre of a reductive group slightly differently, with the combinatorial building effectively discarding it, as @UriBader points out, while the CLT spherical building of a reductive group with (split-)rank-$$d$$ centre is the $$d$$-fold suspension of the spherical building of its derived group.