Let $G=GL_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$. We can write the infinte Grassmannian as a colimit of the finite Grassmannians $Gr(k,n)$, which are honest algebraic varieties, that have the following nice properties:
1) They admit an affine cell paving, the Bruhat stratification.
2) The intersection cohomology complexes of the strata have good vanishing properties: They are parity sheaves, meaning that their restriction to smaller strata have cohomology only in every second degree.
3) On top of that, when "interpreting" the intersection cohomology complexes as say mixed $\mathbb Q_l$ sheaves they have good purity properties: Their restrictions to smaller strata are still pure and even direct sums of shifted Tate twists.
Now my question is the following: If we replace $GL_k(\mathbb C)$ by another connected reductive affine complex algebraic group $G$, are there still algebraic varieties approximating $BG$ such that (some of) the above properties hold?