Equivariant Fredholm modules for the full quantum flag manifold of SUq(3)

We introduce C∗-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct SLq(3, C)-equivariant Fredholm modules for the full quantum flag manifold Xq = SUq(3)/T of SUq(3), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold Xq satisfies Poincar´e duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to SUq(3)