Speaker
Description
The classical Helgason conjecture claimed that the Poisson transform isomorphically maps the space of distributions on the Furstenberg boundary G/P on a semisimple Lie group G to the space of joint eigenfunctions on its symmetric space G/K, as proven in the ‘70s by Kashiwara et al. In a related scenario, Knapp-Wallach studied the Poisson transform infinitesimally intertwining certain non-unitary principal series representations with discrete series representations. In this talk, we will show how the Heisenberg calculus give quantitative control on Sobolev mapping properties of Knapp-Wallach’s Poisson transform for groups of real rank one. Along the way we prove that this Poisson transform is compatible with smooth functions on the Furstenberg compactification up to compact operators, a result that constituted the last missing piece in Julg’s program for the Baum-Connes conjecture for subgroups of real rank one groups. Joint work with Heiko Gimperlein
