Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions about the equivariant cohomology ring $H^\*\_G(\mathbb{P}(V),\mathbb{Q})$.


1. Under what conditions on the representation $V$ the ring $H^\*\_G(\mathbb{P}(V),\mathbb{Q})$ is an integer domain?
2. The second question is more delicate. Suppose that $i\colon X \to \mathbb{P}(V)$ is a closed $G$-equivariant embedding of a $G$-submanifold $X$ (for example, $X$ is a closed $G$-orbit in $\mathbb{P}(V)$). Denote by $i\_!1\in H^\*\_G(\mathbb{P}(V),\mathbb{Q})$ the image of $1\in H^0\_G(X,\mathbb{Q})$ under the equivariant pushforward map $i\_!\colon H^\*\_G(X,\mathbb{Q}) \to H^\*\_G(\mathbb{P}(V),\mathbb{Q})$. I would like to know under what conditions the cohomology class $i\_!1$ is not a zero divisor in the ring $H^\*\_G(\mathbb{P}(V),\mathbb{Q})$.



Equivariant cohomology ring is an integer domain
It is very rare for these rings to be integral domains. To see this, put 
$$ f\_V(t)=\sum\_kc\_k(V)t^{\dim(V)-k} \in H^\*(BG)[t]. $$
(All cohomology here has rational coefficients.) 
It is then standard that $H\_G^\*(PV)=H^\*(BG)[x]/f\_V(x)$, and also that $f\_{V\oplus W}(t)=f\_V(t)f\_W(t)$. From this it is clear that $H\_G^\*(PV)$ can only be a domain if $V$ is irreducible. However, irreducibility is far from being sufficient. To see this, let $T$ be a maximal torus with Weyl group $W$. By a "fake representation" of $G$ I will mean a $W$-invariant representation of $T$. The $W$-orbits in $T^\*$ give a basis over $\mathbb{N}$ for the fake representations. Because $H^\*(BG)=H^\*(BT)^W$, a fake representation $V$ still has an associated polynomial $f\_V(t)\in H^\*(BG)[t]$. Each fake representation is the restriction of a virtual representation of $G$, but not usually a genuine representation of $G$. If a genuine representation $V$ can be decomposed nontrivially as a sum of fake representations, then we get a factorisation of $f\_V(t)$, showing that $H^\*\_G(PV)$ is not a domain. I think it is almost always possible to find such a fake decomposition. To find the list of exceptions is probably an easy exercise for someone more fluent in Lie theory than I am.


