Geometric and Microlocal Study of Automorphic Periods

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In harmonic analysis, one represents functions on a space as a superposition of waves with varying frequencies. In number theory and the Langlands program, one is interested in functions on certain homogeneous spaces, where the "waves" are special eigenfunctions of Laplacians and Hecke operators, called automorphic forms. Some of the most mysterious and important invariants of the automorphic forms are the L-functions, a vast class of generalizations of the Riemann zeta function. A significant, but not well-understood, principle is that their superposition often represents a function that can be described independently, in terms of what are known as spherical varieties that give rise to a distribution called the period distribution. The amplitudes of the spectral decomposition of this distribution turn out to be special values of L-functions. The project will investigate conjectural connections between period distributions and L-functions using ideas of quantization (whose roots lie in mathematical physics). The PI also plans yearly meetings to train students and postdocs on the topics related to this proposal. According to the visionary program developed since the '60s by Abel Prize recipient Robert P. Langlands, L-functions should be understood as invariants of automorphic representations; those are the "eigenfrequencies" of "arithmetic manifolds", or else the representations of a (reductive) Lie group G, and of its algebra of Hecke operators, which appear as functions on a quotient L\G, where L is an arithmetic lattice. The precise incarnation of L-functions in this setting is by means of certain distributions called "periods", which the PI and others have studied and organized into a coherent theory in recent years. The present award aims to utilize ideas of symplectic geometry in the study of these periods. Among other goals, this project will study: (1) the duality between periods and L-functions as a duality between Hamiltonian spaces for the group and its dual group (building up on recent work with Ben-Zvi and Venkatesh); (2) the local spectrum of spherical varieties by combining the relative trace formula of Waldspurger with the geometry of the moment map studied by Knop; (3) the "transfer operators" of functoriality, in the spirit of Langlands' "beyond endoscopy", between the relative trace formulas of different groups and spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.