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Mathematics, Number Theory, math.NT,High Energy Physics - Theory, hep-th,Mathematics, Representation Theory, math.RT
Abstract:
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of Langlands' constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker vector associated to unramified automorphic representations
of G(Q_p). Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
introduce also some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In addition, we
explain how the classical theory of Hecke operators fits into the modern theory
of automorphic representations of adelic groups, thereby providing a connection
with some key elements in the Langlands program, such as the Langlands dual
group LG and automorphic L-functions. Our treatise concludes with a detailed
list of interesting open questions and pointers to additional topics where
automorphic forms occur in string theory.