TITLE: Integral cohomology of $G/N(T)$
QUESTION [14 upvotes]: 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 $G=SU(n)$.

REPLY [3 votes]: Here are some comments about how this calculation can be organised.
First, let $T(n)$ be the standard maximal torus in $U(n)$, and let $N(n)$ be its normaliser, so $W(n)=N(n)/T(n)$ is the symmetric group.  Put $F(n)=U(n)/T(n)$ and $E(n)=F(n)/W(n)=U(n)/N(n)$, so we are interested in the (co)homology of $E(n)$.  We know that $H^*(E(n);\mathbb{Q})=\mathbb{Q}$.  I think it will be best to study $H_*(E(n);\mathbb{Z}/p)$ (where $p$ is prime) and recover information about $H^*(E(n);\mathbb{Z})$ at the end, using the universal coefficient theorems.  From now on I'll write $H_*(X)$ for $H_*(X;\mathbb{Z}/p)$.  It will also be best to study all the groups $H_*(E(n))$ together as a bigraded object $H_*(E(*))$.
Note that there are sum and tensor product operations $\oplus\colon U(n)\times U(m)\to U(n+m)$ and $\otimes\colon U(n)\times U(m)\to U(nm)$.  These are group homomorphisms, so they induce maps $BU(n)\times BU(m)\to BU(n+m)$ and $BU(n)\times BU(m)\to BU(nm)$.  They preserve monomial matrices, so they give maps $BN(n)\times BN(m)\to BN(n+m)$ and $BN(n)\times BN(m)\to BN(nm)$.  We can regard $E(n)$ as the fibre of the map $BN(n)\to BU(n)$, so we also get maps $E(n)\times E(m)\to E(n+m)$ and $E(n)\times E(m)\to E(nm)$.  This will give two different products on the object $H_*(E(*))$, which will make it into a Hopf ring.  Also, using block matrices we get a standard inclusion $\Sigma_n\wr U(m)^n\to U(nm)$.  This gives an action of an $E_\infty$ operad on $BU(*)$ and $BN(*)$ and $E(*)$, and so gives Dyer-Lashof operations on the corresponding homology groups.
Now $BN(*)$ is essentially the free $E_\infty$ space generated by $BN(1)=\mathbb{C}P^\infty$, which implies that $H_*(BN(*))$ is the free commutative algebra with Dyer-Lashof operations generated by $H_*(\mathbb{C}P^\infty)$.  On the other hand, $H_*(BU(*))$ is just the free commutative algebra generated by $H_*(\mathbb{C}P^\infty)$.  This implies that the map $H_*(BN(*))\to H_*(BU(*))$ is surjective (which can also be seen using the Becker-Gottlieb transfer map).  Note also that the Dyer-Lashof operations in $H_*(BU(*))$ were worked out by Priddy.  It seems likely that $H_*(E(*))$ is just the kernel (in the Hopf algebra sense) of the map $H_*(BN(*))\to H_*(BU(*))$.  Some care is needed here because the relevant Hopf algebras do not have antipodes; they correspond to commutative monoids (rather than groups) in the category of coalgebras.  Anyway, the message is that we expect that $H_*(E(*))$ will have structure closely related to that of the Dyer-Lashof algebra, and of similar size and complexity.
So far we have not used the tensor product maps $U(n)\times U(m)\to U(nm)$, or equivalently, the second product in the Hopf ring structure.  It is a theorem of Paul Turner that when $p=2$, the object $H_*(BW(*))$ has a simple presentation as a Hopf ring, which is in some ways more tractable than the presentation using Dyer-Lashof operations.  I expect that there is a similar story for $H_*(BN(*))$, although I do not know whether that has ever been written down.  It is also straightforward to write down the Hopf ring structure on $H_*(BU(*))$, and this may well give a nice description of $H_*(E(*))$.  I am not sure whether a similar approach would work for $p>2$.