Our goal will be to explain how enhanced Langlands parameters can be used to obtain a finer and more precise conjectural formula than the Baum-Connes conjecture alone for the K-theory of the reduced $C^*$-algebra of a reductive $p$-adic group.

We will provide the definition and simple examples of these parameters in the talk, and we will state a "Galois version" of the ABPS conjecture, which is the key tool for stating the formula.

This is joint works with Paul Baum, Ahmed Moussaoui, Roger Plymen and Maarten Solleveld.