Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^\*(G/T,\mathbb{Q})$ is the regular representation of $W$ induced by the $W$-action on $G/T$, and hence $H^\*(G/N(T), \mathbb{Q})\cong H^\*(G/T, \mathbb{Q})^W\cong\mathbb{Q}$. I would like to know whether the integral cohomology $H^\*(G/N(T), \mathbb{Z})$ has been known before. I am particularly interested in the case $G=SU(n)$.



Integral cohomology of $G/N(T)$
There are people on MO more knowledgeable about this topic than I, but since it has been a few days, I will say a few words.


In this case there is a "third isomorphism" of spaces:


**Proposition** $G/N(T)\cong \mathcal{F}/W$


**Proof** The flag variety $\mathcal{F}\cong G/T\cong G\_{\mathbb{C}}/B$ and $W=N(T)/T$ is the Weyl group. Define $\varphi:G/T\to G/N(T)$ by $gT\mapsto gN(T)$. This is clearly a well-defined continuous surjection. The group $W$ acts on the right of $G/T$ (and $G/N(T)$) by $gT\cdot nT=gTnT=gnT$ and $\varphi$ is $W$-equivariant. So $\varphi$ descends to a map $\varphi:\mathcal{F}/W\to G/N(T)$ which is easily seen to be injective, and hence a homeomorphism (since $\mathcal{F}/W$ is compact and $G/N(T)$ is Hausdorff). $\Box$


As shown [here](https://mathoverflow.net/q/105110/12218) there is an inclusion $H^k(\mathcal{F}/W;\mathbb{Z})\hookrightarrow H^k(\mathcal{F};\mathbb{Z})^W$ modulo torsion. Since you know $H^\*(\mathcal{F}/W;\mathbb{Q})\cong H^\*(\mathcal{F};\mathbb{Q})^W$ (by a result in Bredon's book on Transformation Groups; page 120) and the latter is $\mathbb{Q}$ (by Borel's 1953 Annals paper, page 193) you know the free parts since $\mathrm{rk} \ H^k(\mathcal{F}/W;\mathbb{Z})=\mathrm{rk}\ H^k(\mathcal{F};\mathbb{Z})^W.$


That leaves torsion. I spoke with an expert in Shubert varieties who told me that (a) they did not know the answer or a reference to this, and (b) they expected complicated torsion.


So I decided to do an example. First note that $W$ acts freely on $\mathcal{F}$ since if $gwT=gT$, then $w\in T$. Thus, $\mathcal{F}/W$ is a smooth manifold. 


**Example:** $\mathrm{SU}(2)/T\cong S^2$ with $W=\mathbb{Z}/2\mathbb{Z}$ acting freely, so the quotient must be $\mathbb{RP}^2$. That gives torsion in $H^2(\mathcal{F}/W;\mathbb{Z})$ at the first opportunity.


More generally, when $G$ is simply connected $G/T$ is simply connected (by path lifting), and so $H\_1(\mathcal{F}/W;\mathbb{Z})\cong W/[W,W]$. This can be used with Poincaré duality (or a [non-orientable version](https://mathoverflow.net/a/61202/12218)) to garner some further information (in particular, lots of torsion as $W$ gets larger).


