  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.
