Eisenstein series and automorphic representations

Abstract

Note to the readerThis book has grown out of our endeavour to understand the theory of automorphic representations and the structure of Fourier expansions of automorphic forms with a particular emphasis on adelic methods and Eisenstein series. Our intention is also to open a path of communication between mathematicians and physicists, in particular string theorists, interested in these topics. Most of the results presented herein exist already in the literature and we benefitted greatly from [66, 98, 183-185, 297, 389, 405, 406]; our exposition differs, however, at places from the standard one. A few new results and examples are included as well; in particular, we provide many techniques for working out aspects of the Fourier expansion of Eisenstein series. We would be very grateful to learn of any omissions and mistakes that we have made unintentionally.Sections that are more advanced or explore topics beyond the main focus of this treatise are marked with an asterisk. discussions with colleagues both from the mathematics community and from the string theory community. We are especially indebted to

D6R4 curvature corrections, modular graph functions and Poincaré series

“…The trivial case ψ = 1 represents the constant term of f with respect to U ; in physics applications this comprises typically all perturbative contributions in some modulus. 4 The differential equation (2.6) for ϕ also implies that ϕ is not an automorphic function in the standard sense [35]. Standard automorphic functions are what is called Z(g)-finite, where Z(g) is the center of the universal enveloping algebra of the Lie algebra g of E11−D.…”

“…5 For non-abelian U , one only has to include the generators associated with the abelianisation [U, U ]\U . The Fourier expansion will then be incomplete and has to be refined using also non-abelian Fourier coefficients [35,49,50].…”

“…and E (1,0) that correspond to the 1 2 -BPS R 4 and 1 4 -BPS D 4 R 4 curvature corrections, respectively. Their solutions have been studied in great detail and are given by (linear combinations of) Eisenstein series, see for instance [14,15,17,25,35] for summaries or section 2.2 below. By contrast, equation (1.3c) for E (D) (0,1) (that corresponds to the 1 8 -BPS correction D 6 R 4 ) is of a qualitatively different nature in that it always contains a non-linear source term given by −(E…”

“…The structure of the inhomogeneous Laplace equation (1.3c) indicates that it will not be solved by an automorphic function of the standard type as these functions are required to be finite under all E 11−D -invariant differential operators, see for instance [35]. The generalised class of E 11−D (Z)-invariant functions on E 11−D to which the solution E (D) (0,1) belongs has not been identified abstractly.…”

A general framework of automorphic inflation

“…The classical series were generalized to the automorphic framework by several authors in the 1950s and 60s, in particular Selberg and Langlands, with a more detailed account published by Harish-Chandra [23]. An extensive account of Eisenstein series in the adelic framework can be found in [33]. While the adelic language does have advantages for certain questions, and much of the modern literature on automorphic forms and representations is written in it [34], this formulation is not always the best approach because it can obscure the geometric structures present in certain physical problems.…”