@Donu: could you give me at least one reference about that spectral sequence? thank you

OK, perfect!.... er... I mean, thank you, that would be awesome!

@DonuArapura, what does "perfect" mean for the stratification? and how does the situation get better in that case? and BTW what is the spectral sequence that abuts to that? (too many questions, I know)

I have edited the question to make a better link with the answers.

Yes, you are right, I am quite confused. The reduction modulo 2 of $E_8$ must have a quadratic form with values in $\mathbb{F}_2$ hence there is a non trivial map from $W(E_8)$ to the orthogonal group $O^+(8,\mathbb{F}_2)$ (with kernel $\pm 1$). So what is the relation with $\mathbb{F}_2^8$?