15:00 - 15:55:   Raphaël BEUZART-PLESSIS (Aix-Marseille Université and CNRS)
Title:Multipliers and isolation of the cuspidal spectrum by convolution operators
 
Abstract:
Let G be a real reductive algebraic group and Γ be an arithmetic lattice of G. In this talk, I will explain how to generalize a construction of Lindenstrauss-Venkatesh giving rise to certain operators on L²(Γ\ G) with image in the cuspidal subspace. These operators can be written, in the adelic setting, as combinations of convolution operators at Archimedean places and p-adic places (Hecke operators). A crucial ingredient of the proof is the existence of sufficiently many multipliers of G acting on the space of smooth functions with rapid decay (but not necessarily K-finite). Time permitting, I will also describe one application of this construction to the global Gan-Gross-Prasad conjecture for unitary groups. This talk is based on joint work with Yifeng Liu, Wei Zhang and Xinwen Zhu.

Video: [mp4]

16:15 - 17:10:   Erik VAN DEN BAN (University of Utrecht)
Title: The Harish-Chandra transform for Whittaker functions
 
Abstract:
I will discuss the role of the descent transform in Harish-Chandra's approach to the Plancherel formula for Whittaker functions, presented in the posthumous volume 5 of his collected works (Springer 2018). At an earlier occasion I explained how the proof of the Plancherel theorem can be completed by using a Paley-Wiener shift technique. In the present talk I will explain how the proof can be completed in a more straightforward way, by using a suitable result on wave packets of Whittaker functions.

Video: [mp4]