Bessel functions on higher-rank groups

Classical Bessel functions arise as solutions to certain differential equations and appear throughout mathematics and the sciences. There are numerous books on their properties and behaviors and descriptions as integrals, series, and so on. They appear in number theory through formulas of Voronoi, Petersson and Kuznetsov. In particular, the formula of Kuznetsov, also known as a relative trace formula, gives connections between interesting objects in analysis and algebra/geometry. This research studies the generalizations of Kuznetsov's formula and the generalized Bessel functions that appear in such formulas, with the goal of building a body of knowledge similar to what is known about the classical Bessel functions. This will allow analytic number theorists to explore the applications of these new Kuznetsov-type formulas. Part of the funding for the project will be used to support undergraduate and master's-level research in analytic number theory. While the Kuznetsov formula for SL(2) has been the subject of intense research for the past 40 years, much less is known about its generalizations to reductive groups. These generalizations relate the exponential sums occurring in the Fourier coefficients of Poincare series to the Fourier coefficients of automorphic forms and the integral transforms in these generalized Kuznetsov formulas can, conjecturally, be expressed as kernel integral transforms; the kernels are generalized Bessel functions. This research aims to extend the study of Bessel functions to groups such as GL(n) and GSp(n). The immediate goals are to study their differential equations, integral representations and give basic asymptotics of their integral transforms in low-rank groups such as GL(4) and GSp(4). Long-term goals for the project include extending these results to arbitrary rank and applications such as the arithmetically-weighted Weyl law. Smaller, associated projects on GL(3) include studying the Poincare series occurring in the GL(3) Fourier expansion, application of the Kuznetsov formula to the supremum norm problem as well as the relationship between vector sup norms and scalar sup norms for Maass forms with weight. This project is jointly funded by Algebra and Number Theory program, and the Established Program to Stimulate Competitive Research (EPSCoR). 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.