An equivariant Poincaré duality for proper cocompact actions by matrix groups

Let $G$ be a linear Lie group acting properly on $G$-$\mathrm{spin}^c$ manifold $M$ with compact quotient. We give short proof that Poincaré duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional $G$-vector bundles, and $K$-homology through the geometric model Baum Douglas.