Automorphic Forms on Loop Groups

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The aim of this proposal is to study the link between two important mathematical developments from the late 1960s, both by Canadian mathematicians. The first of these was the visionary program of Robert Langlands to apply the techniques of harmonic analysis to the central questions in number theory. The carriers of this number theoretic information are automorphic L-functions, and a cornerstone of this program has been the study of the analytic properties of these functions. Since its inception, the Langlands program has grown to become nearly ubiquitous in mathematics, having both influenced and been influenced by important developments in representation theory, algebraic geometry, string theory, and even algebraic topology. The second of these developments was the introduction by Victor Kac and Robert Moody of a family of infinite-dimensional symmetries generalizing the classical Lie theory of the century preceding them. Although infinite-dimensional in nature, these new objects were shown to possess many of the same structural features as their finite-dimensional counterparts. Moreover, they often also exhibited surprising and far reaching connections with other areas of mathematics. Nowhere has this been more noticed than in the ‘simplest’ class of infinite-dimensional objects introduced by Kac and Moody, the so called loop groups and loop algebras (or affine Kac-Moody groups and algebras). Spurred in parts by the theory of finite simple groups and also by developments in mathematical physics, loop algebras and groups were connected in the 1980s with the central questions in algebraic combinatorics, the theory of vertex operators, and the theory of quantum groups. The goal of this proposal is to enlarge the range of the Langlands program by considering it not just for finite-dimensional groups, but also for infinite-dimensional loop groups. A central facet of the Langlands philosophy is that L-functions (and the number theoretical questions which they encode) on even the smallest of groups should not be studied in isolation. Rather, one needs to consider them alongside analogous objects on larger and often quite different groups. Put together, this information gives powerful insights into the original question, and the more groups one can consider, the more refined the insight into the original problem. A vehicle for moving from smaller to larger groups is the Eisenstein series construction. It's counterpart, the constant term (or more generally the Fourier coefficient) construction moves from larger to smaller groups. Langlands and Shahidi have developed the machinery to analyze automorphic L-functions by composing these two constructions. Namely, they first form an Eisenstein series, then study it using tools from harmonic analysis, and finally return to the original setting using the constant term construction. The efficiency of this construction relies on the fact the Eisenstein series is amenable to study via spectral techniques, not available in the original setting. In practice, this paradigm has produced the strongest known results in a variety of celebrated questions in analytic number theory. To get the Langlands-Shahidi method started, one needs a rich supply of larger groups on which to consider Eisenstein series. There are a limited number of such groups within a finite-dimensional context, the information from most of which have already been gleaned. However, introducing infinite-dimensional groups into the picture vastly broadens the scope of the method. This proposal aims to develop the machinery to incorporate these infinite-dimensional groups into the Langlands program, and hence into the arsenal of tools available attack the central questions of analytic number theory.