On the Connes-Kasparov isomorphism, II
The Vogan classification of essential components in the tempered dual

Pierre Clare, Nigel Higson, Yanli Song

Abstract: This is the second of two papers dedicated to the computation of the reduced $\mathrm{C}^{*}$-algebra of a connected, linear, real reductive group up to $\mathrm{C}^{*}$-algebraic Morita equivalence, and the verification of the Connes--Kasparov conjecture in operator $K$-theory for these groups. In Part I we presented the Morita equivalence and the Connes--Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual. In fact we shall go further by giving a complete representation-theoretic description and parametrization, in Vogan's terms, of the essential components of the tempered dual, which carry the $K$-theory of the tempered dual.