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 $G$? 



Tits building of a linear algebraic group
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) .


