This book focuses on a conjectural class of zeta integrals which arose from a program born in the work of Braverman and Kazhdan around the year 2000, the eventual goal being to prove the analytic continuation and functional equation of automorphic L-functions.

Developing a general framework that could accommodate Schwartz spaces and the corresponding zeta integrals, the author establishes a formalism, states desiderata and conjectures, draws implications from these assumptions, and shows how known examples fit into this framework, supporting Sakellaridis' vision of the subject. The collected results, both old and new, and the included extensive bibliography, will be valuable to anyone who wishes to understand this program, and to those who are already working on it and want to overcome certain frequently occurring technical difficulties.

Wen-Wei Li received his Ph.D in Mathematics from Université Paris-Diderot in 2011. He is currently a Professor of the Beijing International Center of Mathematical Research, Peking University. His research is mainly focused on the theory of representation and automorphic forms.

Introduction.- Geometric Background.- Analytic Background.- Schwartz Spaces and Zeta Integrals.- Convergence of Some Zeta Integrals.- Prehomogeneous Vector Spaces.- The Doubling Method.- Speculation on the Global Integrals.