Representations of Kac-Moody Groups and Applications to Automorphic Forms

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This proposal lies at the intersection of representation theory and number theory and concerns a class of infinite-dimensional groups known as Kac-Moody groups. We focus both on developing tools necessary for concrete applications to well-known questions in number theory, and also on introducing new constructions in infinite-dimensional representation theory. Applications to Number Theory The theory of Eisenstein series on loop groups (the simplest of the infinite dimensional Kac-Moody groups) comes in two related flavours-- number fields and the function fields. In the function field setting, a precise proposal was put forward in joint work with H. Garland and S.D. Miller (building on work of A. Braverman and D. Kazhdan) relating automorphic L-functions from finite dimensions to infinite-dimensions. To carry this project to fruition, one needs to analytically continue a new object which appears-the so called negative cuspidal Eisenstein series. Our understanding in the number field setting lags behind the function field setting because of the absence of a theory of spherical functions for real loop groups. Inspired by techniques from probability theory and quantum groups, we propose to study a Gaussian measure on these groups and extract from them concrete formulas for spherical functions and their limits. In recent work with A. Puskas, we have introduced a metaplectic cover of any simply-connected Kac-Moody group and studied local Whittaker functions on it. It has long been expected that global Fourier coefficients of Eisenstein series on such a cover should be linked to very concrete questions about moments of Dirichlet L-functions. Having made the first links in this direction, we propose to continue these studies. Generalizing the work of my PhD thesis (which was in the function field setting), we propose a geometric interpretation of arithmetic quotients of loop groups over number fields. We expect this to be connected to a recent Arakelov theory for formal surfaces (due to J.B.-Bost) and we plan to apply our interpretation to estimate for the number of rational points of certain moduli spaces of bundles on an arithmetic surface. New Techniques in Representation Theory of Kac-Moody Groups Based on our previous joint work with A. Braverman and D. Kazhdan on Hecke algebras, we now propose to investigate the notion of cuspidal representations, the building blocks of all representations, and study their L-functions. Together with D. Muthiah, we also propose to study a notion of double affine Kazhdan-Lusztig functions. Physicists interested in low-energy limits of certain string theories were led to studying "minimal" automorphic representations on certain exceptional Lie groups. They also postulated the existence, based on remarkable calculations, of such representations for the infinite-dimensional groups. We propose to develop aspects of a Kirillov-Duflo orbital theory in infinite-dimensions to mathematically construct these.