Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology.
My question: In what sense can we have a duality between the equivariant homology and cohomology, in analogue with the Poincare duality between the ordinary homology and cohomology of $X$? In particular, the degree of a nontrivial equivariant (co)homology class could exceed the dimension of the manifold $X$. Then in such case, what does the dual mean, geometrically?