You're certainly right about H^1 being nontrivial if I is principal; I forgot to mention that. However I'm really interested about the 'internal structure' so to speak: the cohomology H^i when i is less than the number of generators. Thank you for your reference though!

Dear LSpice, you are quite right in that here the $P_i$ are not minimal parabolic but rather the smallest standard parabolic subgroups such that $Lie(P_i)$ contains the root space spanned by $e_{-\alpha_i}$; this was an error in my terminology.

Okay, will do. Thank you very much for your reference and comments.

How does this match with the statement in Kazhdan-Lusztig "Fixed Point Varieties on Affine Flag Manifolds" , where they say that the set of conjugacy classes of maximal tori in $G(\mathcal{K})$ correspond to $H^1(\Gamma, N(\overline{F}))$? My impression is that they were working with the short exact sequence $1 \rightarrow T \rightarrow N_G(T) \rightarrow W \rightarrow 1$; are you using Galois Cohomology for $1 \rightarrow N_G(T) \rightarrow G \rightarrow G/(N_G(T))$, where we have the scheme parametrizing tori on the right? I'm a little confused

Dear Mikhail, this looks like the closest reference I could possibly expect. Thank you very much.

I would also be delighted if anyone felt like there was a good reference (even if it only deals with $\mathbb{C}/\mathbb{R}$) for this material.