Let be $X$ a maximal torus in a Lie group $G$. I'd like to calculate the cohomology $H^{*}(G/N(T))$. I know that it is trivial in odd degree and the basefield is even but I haven't a basic method to do this. Is there a simple method of calculation when the gruop $G$ is the unitary gruop $U(n)$?
If you take $G=U(n)$, then $G/T$ is the flag variety of $GL_n({\mathbb C})$. Hence its cohomology is given by Bruhat cells and as a representation of the Weyl group $W=N(T)/T)$ acting on the right on $G/T$, it is the regular representation (there are no fixed points for $W$!Use Lefschetz). Therefore, the space of invariant elements is one dimensional and is given by $H^0(G/T)$. The same method applies to any compact connected LIe group $G$ and $T$ is a maximal torus. For other groups, you need, I think, to know what the torus looks like (i.e. if it is split, or nonsplit etc). 

