Analysis and Probability seminar

Abstract: We discuss Poisson transforms for semisimple Lie groups. Our main results concern Knapp-Wallach’s Szegö maps from certain non-unitary principal series representations to discrete series representations, and hold in real rank one. We prove sharp mapping properties for the Szegö maps and compactness of commutators with functions on the Furstenberg boundary. The results are motivated by Julg’s program to prove the Baum-Connes conjecture with coefficients in real rank one, and prove the remaining open conjecture of Julg. Based on joint work with Heiko Gimperlein.