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

Abstract

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.

Complete rewrite. Paper has been rewritten to take advantage of the paper "Products of Longitudinal Pseudodifferential Operators on Flag Varieties" (at http://dx.doi.org/10.1016/j.jfa.2009.10.007). Also added a result concerning order zero longitudinal pseudodifferential operators tangent to the fibrations of the flag variety of SL(3,C) -- Theorem 3.18