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

TL;DR

This paper constructs a $G$-equivariant $K$-homology element for $SL(3,C)$ using the BGG complex, providing explicit splittings in representation rings and a new model for the gamma element, leveraging harmonic analysis techniques.

Contribution

It introduces an explicit construction of a $K$-homology element for $SL(3,C)$ and offers a novel model for the gamma element through BGG complexes and harmonic analysis.

Findings

Explicit splitting of the restriction map from $R(G)$ to $R(K)$

New model for the gamma element of $SL(3,C)$

Application of harmonic analysis on the flag variety

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.