QUANTUM DYNAMICS BANACH CENTER PUBLICATIONS, VOLUME 120 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2020 SOME ASPECTS OF THE GEOMETRIC STRUCTURE OF THE SMOOTH DUAL OF p-ADIC REDUCTIVE GROUPS ANNE-MARIE AUBERT CNRS, Sorbonne Université, Université de Paris Institut de Mathématiques de Jussieu – Paris Rive Gauche F-75005 Paris, France ORCID: 0000-0002-9613-9140 E-mail: anne-marie.aubert@imj-prg.fr Abstract. We review the ABPS Conjecture in the case of a split p-adic reductive group G, its links with the Langlands correspondence for G, and how it can be used in order to provide a conjectural description for K*(C * red (G)). 1. Introduction and a few memories. The first time I heard the name of Paul Baum was in a talk given by Vincent Lafforgue at the École Normale Supérieure on the “Baum– Connes conjecture” in the end of the 90’s. A few years later, while I was already working with Roger Plymen, I had the opportunity to start collaborating with Paul also. A series of exchanges of emails and files with Paul and Roger led to two papers [ABP1], [ABP2] on the very early stages of what is now known as the “ABPS conjecture”. A version of the conjecture, that was stated and studied jointly with Paul, Roger and Maarten Solleveld, is recalled in Section 3. I met Paul for the first time at the Institut Henri Poincaré (IHP) in Paris, on his way to visiting the Université Clermont Auvergne. I had a very interesting mathematical discussion with Paul, and he talked also in French with my Mother and told her that he travelled several times with his own Mother to mathematical conferences. We met again at several occasions, notably at three “birthday conferences”: the one in honor of Alain Connes in 2007 in Paris, the one in honor of Roger the year after in Manchester, and the “Conference on Geometry, Representation Theory and the Baum– Connes Conjecture” in 2016 at the Fields Institute in Toronto, whose aim was also to celebrate the 80th birthday of Paul. I was very glad and honored in giving a talk at 2010 Mathematics Subject Classification: Primary 20C08; Secondary 14F43, 20G20. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc120-10 [135] c  Instytut Matematyczny PAN, 2020