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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 $G$?

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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) .

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  • $\begingroup$ 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). $\endgroup$ – Kiu May 6 '18 at 1:11
  • $\begingroup$ 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. $\endgroup$ – Kiu May 6 '18 at 1:12
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    $\begingroup$ @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. $\endgroup$ – Uri Bader May 6 '18 at 8:52
  • $\begingroup$ 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". $\endgroup$ – Kiu May 6 '18 at 15:11
  • $\begingroup$ No offense was intended. Sorry for any confusion. I accepted you answer. $\endgroup$ – Kiu May 7 '18 at 14:51

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