TITLE: Nice algebraic approximations of classifying spaces
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: I think there is some subtlety in the case of the symplectic group that has not yet been discussed in the two other answers. 
Embedding a symplectic bundle into a trivial symplectic bundle and constructing a classifying space from that actually means we are looking at the symplectic Grassmannian as classifying space. To recall what the symplectic Grassmannian is, let $V$ be a vector space of dimension $2n$, equipped with a symplectic form. The symplectic Grassmannian  $\operatorname{HGr}(2r,2n)$ is the subset of the Grassmannian $\operatorname{Gr}(2r,2n)$ consisting of those $2r$-dimensional subspaces of $V$ on which the symplectic form has non-degenerate restriction. It can then be identified with the quotient $\operatorname{Sp}_{2n}/(\operatorname{Sp}_{2r}\times\operatorname{Sp}_{2n-2r})$. This, however, is not a flag variety, it is affine and not projective. 
The geometry of these symplectic Grassmannians is more subtle than of the ordinary Grassmannians for the general linear group. 
In particular, they do not admit a paving with affines; they admit a paving with quasi-affines which are $\mathbb{A}^1$-contractible. This failure of the existence of a paving with affine spaces can already be seen at the level of motives for $\operatorname{HGr}(2,2n+2)$. The motives of these "quaternionic projective spaces" are 
$$
\operatorname{M}(\operatorname{HGr}(2,2n+2))\cong \bigoplus_{i=0}^n\mathbb{Q}(2i)[4i].
$$
The corresponding (but too complicated to write out here) statements for the other symplectic Grassmannians exactly fit  the topological fact that Pontryagin classes live in dimensions which are multiples of $4$.
If there was a paving with affines, some motives $\mathbb{Q}(i)[2i]$ for $i$ odd would have to appear. Somehow, geometrically, what prevents the paving to be by affine spaces (instead of just contractible quasi-affine varieties) in a motive as above is that the cells have to have a boundary which is of codimension $\geq 2$, which can not happen for affine spaces.
Anyway, all of the above can be found and is beautifully discussed in the paper by Panin and Walter on "Quaternionic Grassmannians and Pontryagin classes in algebraic geometry".
The result is that (1) is not true for symplectic Grassmannians. As far as I see, this does not really cause problems with (2) and (3); only the arguments have to be made a bit more carefully, because the stratification is not by affine spaces, but by more complicated (but still contractible) quasi-affine varieties.