Abstract: Conjectures of Braverman-Kazhdan, Lafforgue, Ng\^{o} and Sakellaridis suggest that every affine spherical variety admits a generalized Poisson summation formula. We refer to this conjecture as the Poisson summation conjecture. The Poisson summation conjecture implies the functional equation and meromorphic continuation for fairly general Langlands $L$-functions, which by the converse theorem, implies Langlands functoriality in great generality. In collaboration with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie, we constructed a family of period integrals using certain spherical varieties related to Braverman-Kazhdan spaces, which are holomorphic multiples of the triple product $L$-function in a domain that nontrivially intersects the critical strip. If time permits, I'll also talk about some current progress towards the analytic properties by introducing a family of Whittaker inductions to the picture.