On regularization of integrals of matrix coefficients associated to spherical Bessel models
The Gan-Gross-Prasad conjecture relates a special value of an L-function of two cuspidal automorphic representations to the non-vanishing of a certain period. The Ichino-Ikeda conjecture is a refinement of this conjecture. It roughly states that the absolute value of the square of the period in question can be expressed as a product of the special value of the L-function and a product of normalized local periods. However, in order to formulate this conjecture, one needs to assume that the representations in question are tempered everywhere, or else the convergence of the local periods is not guaranteed. The generalized Ramanujan conjecture speculates that the representations in question (cuspidal automorphic representations lying in generic packets) are already tempered everywhere. However, the generalized Ramanujan conjecture is far from being known. In this talk, I will explain how to drop the assumption that the representations are tempered almost everywhere. I will explain how to extend the definition of the normalized local periods for places where the local components are given by principal series representations.