Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

I will talk about how Delorme's techniques used in his proof of the Paley-Wiener theorem can be applied to the Casselman-Wallach Schwartz algebra of rapidly decreasing smooth functions on a real reductive group, and how they can be used to give a sort of structure theorem strong enough to prove a refined version of the Connes-Kasparov isomorphism. If there's time, I will discuss how the Connes-Kasparov isomorphism (in the formulation I use) fits into an "Oka principle" philosophy for Baum-Connes (which started with Bost), and I will discuss further ideas in this direction.