Foundations of the Relative Langlands Program

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The Langlands program is at the heart of modern number theory, and its striking conjectures have dominated much of the field. Its main predictions include a connection (called reciprocity) between the world of diophantine equations and a world of analytic objects called automorphic forms, and surprising relations (called functoriality) within the latter. The present project focuses on the second of these predictions which, as much as it has been studied in recent decades, is lacking a proper understanding (what exactly is the nature of functoriality? what are the objects that we are supposed to compare?) or strategy for its proof (besides the extremely important, but relatively limited instances of "endoscopy" proved with the Arthur-Selberg trace formula). The project will establish the foundations for a generalization of these conjectures -- termed "relative functoriality" -- as follows: the objects to be compared are Schwartz functions on algebro-geometric spaces called "stacks," and ways to compare them will be investigated along the lines of the "beyond endoscopy" ideas set forth by Langlands.More precisely, the algebraic stacks to be considered arise from a pair of spherical varieties with an action of a reductive group G. There is an L-function attached to each of these varieties, generalizing the method of Rankin-Selberg and period integrals; this L-function will be studied building upon earlier work of the PI and others. The relative trace formula of Jacquet will be further developed, generalizing the Arthur-Selberg trace formula, as a distribution on the quotient of these varieties by the diagonal action of G. Starting from low-rank cases, the investigator will examine new, "non-standard" ways of comparing relative trace formulas via integral transforms. Along the way, new results in harmonic analysis on homogeneous spaces will be developed.