The Poincaré duality for an oriented n-manifold M takes the form : $$H^\star(M) \simeq H_c^{n-\star}(M)^\vee.$$ Instead of M take now a real Lie group G. We can basically study it by looking at its topology (i mean by forgetting its group structure), or at its representation theory. These correspond respectively to the modules and comodules over the Hopf algebra $O_G$ ; the studies are dual in that sense.
The topological study makes appear some sheaves of $O_G$-modules, the De Rham cohomology groups $H^\star(G)$ and so the cyclic homology groups $HC_\star(O_G)$. For the group structure study, instead of looking at comodules over $O_G$, we look at modules over the crossed product $\mathbb{C} \rtimes G$ ; we deal with cosheaves of $\mathbb{C} \rtimes G$-modules and the cyclic homology groups $HC_\star(\mathbb{C} \rtimes G)$.
Since the compactly supported homology appearing in the Poincaré duality's statement is a cohomology of cosheaf of $\mathbb{C} \rtimes G$-modules, my question is the following : does the Poincaré duality relate the topological study and the group structure study of a given Lie group ? I mean, does it jump over the gap of "module-comodule over $O_G$" duality ?
