Plancherel and Poisson summation formulae for a family of affine $\Psi$-bundles

Abstract

We prove a Plancherel formula for a family of affine $\Psi$-bundles. As an application, we construct Fourier transforms and asymptotic Schwartz spaces for the family. We then prove the corresponding Poisson summation formula under suitable assumptions. The choice of affine $\Psi$-bundles we consider is motivated by applications to triple product $L$-functions explored in another paper of the authors and Leslie.