Equivariant cohomology ring is an integer domain

Question

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 $H^*_G(\mathbb{P}(V),\mathbb{Q})$.

Answer 1

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.