Banach KK-theory and the Baum-Connes conjecture

The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group $C^{\ast }$-algebras, and a new proof of the classification by Harish-Chandra, the construction by Parthasarathy and the exhaustion by Atiyah and Schmid of the discrete series representations of connected semi-simple Lie groups.