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We introduce $C^*$-algebras associated to the foliation structure of aquantum flag manifold. We use these to construct$SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flagmanifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of theBernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flagmanifold $ X_q $ satisfies Poincar\'e duality in equivariant $ KK $-theory.Moreover, we show that the Baum-Connes conjecture with trivial coefficientsholds for the discrete quantum group dual to $SU_q(3)$.

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