The Bernstein–Gelfand–Gelfand complex and Kasparov theory for SL(3,C)

For G=SL(3,C), we construct an element of G-equivariant analytic K-homology from the Bernstein–Gelfand–Gelfand complex for G. This furnishes an explicit splitting of the restriction map from the Kasparov representation ring R(G) to the representation ring R(K) of its maximal compact subgroup SU(3), and the splitting factors through the equivariant K-homology of the flag variety X of G. In particular, we obtain a new model for the γ-element of G.The construction is made using SU(3)-harmonic analysis associated to the canonical fibrations of X. On this matter, we prove results which demonstrate the compatibility of both the G-action and the order zero longitudinal pseudodifferential operators with the SU(3)-harmonic analysis.