Equivariant Fredholm modules for the full quantum flag manifold of \(\mathrm{SU}_q(3)\) (Q273503)

Summary: We introduce \(C^*\)-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct \(\mathrm{SL}_q(3,\mathbb{C})\)-equivariant Fredholm modules for the full quantum flag manifold \(X_q = \mathrm{SU}_q(3)/T\) of \(\mathrm{SU}_q(3)\), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold \(X_q \) satisfies Poincaré duality in equivariant \(KK\)-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to \(\mathrm{SU}_q(3)\).