Pseudodifferential Operators and the Connes-Kasparov Isomorphism

We compute the K -theory of the C* -category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G . Our result uses the Connes-Kasparov isomorphism for G , and in fact it is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan’s well-known parametrization of the tempered irreducible representations of G with real infinitesimal character. When the reductive group G has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a K -theoretic version of Vogan’s theorem.