Cohomology of root data

Let $(X,\Phi,X^\vee,\Phi^\vee)$ be a semisimple root datum (in the sense of SGAIII), and $W_0$ its (finite) Weyl group. What is known about the cohomology groups $H^n(W_0, X^\vee)$ ?

• You should probably elaborate on your definition, since root data means slightly different things to different people. – Graham Denham Apr 27 '18 at 13:29
• What do you mean by a simple root datum? – Mikhail Borovoi Apr 27 '18 at 13:29
• By that I mean semisimple and indecomposable. Obviously it suffices to consider indecomposable ones, once you restrict to semisimple ones. I reworded it now to make it less confusing. – Nicolas Schmidt Apr 27 '18 at 14:08
• I think that for $G={\rm SL}(n)$ we have $H^1(S_n, X^\vee)\cong\mathbb Z/n \mathbb Z$. – Mikhail Borovoi Apr 28 '18 at 18:04
• @MikhailBorovoi Thanks Mikhail! Am I right to assume that this is due to your own calculations, and that you are not aware of any published results concerning these groups? – Nicolas Schmidt Apr 29 '18 at 14:38