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?



Nice algebraic approximations of classifying spaces
There is good hope for other groups in nice infinite families. $BGl\_k$ is the classifying space of $k$-dimensional vector bundles. $Gr(k,n)$ is the classifying space of $k$-dimensional vector subbundles of an $n$-dimensional trivial vector bundle.


Similarly, $BSP\_{2k}$ is the classifying space of $2k$-dimensional symplectic vector bundles. We can embed each such bundle into a trivial $2n$-dimensional symplectic vector bundle by the same trick one uses for the Grassmanians, and homotope two such embeddings to each other, again using an identical trick. This allows us to write $BSP\_{2k}$ as a limit of flag varieties of $SP\_n$.


We can do the same thing for $O\_n,SL\_n$, and $SO\_n$. For $SL\_n$ we might need to make it noncompact? I am not sure exactly how far one can extend this technique.


The first property is of course a consequence of being a flag variety. I am not so sure about the rest. 


