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 base-field 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 $U(n)$?
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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 non-split etc).
answered Jan 12, 2013 at 7:27
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$\begingroup$ Have you some reference to study how Bruhat decomposition works? How can I use Lefschetz? However, do you know a complete reference in this topic? $\endgroup$ Commented Jan 13, 2013 at 13:51
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$\begingroup$ the space $G/T$ is the same as $G({\mathbb C}/B$ where $B$ is a Borel subgroup of the complexification $G({\mathbb C})$ containing $T$. One is doing Bruhat decomposition for this $B$. This is standard stuff in algebraic groups. The group $N(T)/T$ acts on the {\bf right} on $G/T$ and no element has a fixed point. Hence you can apply Lefschetz fixed point formula to say that this is the regular representation of $W$. Again, this is a well known fact (perhaps old papers of Bott or Borel). $\endgroup$ Commented Jan 13, 2013 at 14:23