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

Let $G=\mathrm{SL}(3,\mathbb{C})$. We construct an element of $G$-equivariant $K$-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, and the splitting factors through the equivariant $K$-homology of the flag variety of $G$. In particular, we obtain a new model for the gamma element of $G$. The proof makes extensive use of earlier results of the author concerning harmonic analysis of longitudinal psuedodifferential operators on the flag variety.