Eisenstein Series on Kac-Moody Groups
and Applications to Physics
November 12 - 20, 2015 /  Rm. 1114, KIAS, Seoul, Korea
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Special Lectures

1) Henry H. Kim (UToronto)
Title: Eisenstein series and Langlands-Shahidi method
Abstract: Eisenstein series play a central role in the theory of automorphic forms. In particular, Langlands-Shahidi method uses Eisenstein series to obtain analytic properties of automorphic L-functions. I will explain how Eisenstein series is used in the applications such as Langlands-Shahidi method, Rankin-Selberg method, spectral decomposition, construction of cusp forms (Ikeda lift).

2) Axel Kleinschmidt (Max Planck--AEI)
Title: Introduction to string theory
Abstract: The basic properties of string theory and string scattering amplitudes will be introduced. Particular emphasis will be given to underlying mathematical structures and the aspects relevant to the occurrence of automorphic forms and representations. These topics include string sigma models, toroidal compactification, duality symmetries, supersymmetry and supergravity.

3) Kyu-Hwan Lee (UConn)
Title: Kac-Moody algebras and their representations
Abstract: After defining Kac-Moody algebras, we will consider their basic structures and integrable representations.

4) Dongwen Liu (Zhejiang)
Title: Construction of Kac-Moody groups
Abstract: We give an introduction to the constructions of Kac-Moody groups, from either a functorial or representation theoretic point of view. The realization in the affine case is more concrete, which is a central extension of the loop group twisted by a degree operator. More details will be given in this case, including in particular a comparison of two different constructions. Group structures such as Bruhat decomposition and Tits system will be also discussed.

5) Manish Patnaik (Alberta)
Title: Properties of Kac-Moody groups
Abstract: In this lecture, Iwasawa decomposition for a general Kac-Moody group and the reduction theory for loop groups will be discussed.

Conference Talks

Speaker: Guillaume Bossard (Ecole Polytechnique)
Title: Exceptional field theory amplitudes
Abstract: Exceptional field theory defines a truncation of string theory to 1/2 BPS states that is manifestly U-duality invariant. The corresponding loop amplitudes are automorphic functions of the string theory moduli. I will explain how exceptional groups (of type E_d) Eisenstein series and generalisations thereof naturally appear in this framework.

Speaker: Dohoon Choi (Korea Aerospace)
Title: Structures for pairs of mock modular forms with the Zagier duality
Abstract: Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called Zagier duality. After the result of Zagier, this type of duality was studied broadly in various viewpoints, including the theory of a mock modular form. In this talk, this problem will be considered with Eichler cohomology theory, especially the supplementary function theory developed by Knopp.

Speaker: Philipp Fleig (Institut des Hautes Études Scientifiques)
Title: Kac-Moody Eisenstein series in string theory
Abstract: The goal of this talk is twofold. In the first part of the talk a concise summary of string theory will be provided with an emphasis on string scattering amplitudes involving (maximal parabolic) Eisenstein series on the E_n exceptional groups. For the second part of the talk we will consider examples of such physically relevant Eisenstein series on Kac-Moody groups, i.e. when n=9,10 and 11, and present some methods allowing for an explicit computation of their Fourier expansions.

Solomon Friedberg (Boston College)
Title: The Whittaker coefficieints of Eisenstein series on covering groups
Abstract: In this talk, I discuss the computation of the Whittaker coefficients of Eisenstein series on covering groups. In particular for maximal parabolic subgroups the computation involves an exponential sum that is expressed in terms of canonical bases, and both the Lusztig data and string data expressions for canonical bases enter into the computation.

Speaker: Stéphane Gaussent (Université Jean Monnet)
Title: Using hovels: Hecke algebras associated with Kac-Moody groups over local fields
Abstract: In this talk, I will explain how to use the hovel to construct two Hecke algebras associated with a Kac-Moody group over a local field. First, I will introduce the spherical Hecke algebra and discuss the Satake isomorphism. Then, the Iwahori-Hecke algebra will come into play and I will explain the Bernstein-Lusztig presentation.

Speaker: Dihua Jiang (Minnesota)
Title: On Discrete Spectrum and Fourier Coefficients for Automorphic Forms
Abstract: Based on the endoscopic classification of Arthur on the discrete spectrum of square-integrable automorphic forms for classical groups, we investigate the global Arthur packets using the notion of Fourier coefficients of automorphic forms associated to nilpotent orbits. This is a report on the work in progress joint with Baiying Liu.

Speaker: Chang Heon Kim (Sungkyunkwan)
Title: Congruences for Hecke eigenvalues in higher level cases
Abstract: In level one case, Guerzhoy defined certain quotient space dual to the space of cusp forms of given weight, developed its properties and applied them to the congruences for Hecke eigenvalues. In this talk, I will extend his result to higher level cases associated to the group generated by the Hecke group and the Fricke involution. This is a joint work with SoYoung Choi.

Speaker: Henry H. Kim (UToronto)
Title: Construction of cusp forms on the exceptional group of type E_7 using Eisenstein series
Abstract: Let G be the connected reductive group of type E_7 and T be the corresponding symmetric domain in C^{27}. Let Gamma be the arithmetic subgroup. For any positive integer k > 10, we will construct a (non-zero) holomorphic cusp form on T of weight 2k with respect to Gamma from a Hecke cusp form of weight 2k-8. We follow Ikeda's idea of using Siegel's Eisenstein series, their Fourier-Jacobi expansions, and the compatible family of Eisenstein series.

Speaker: Axel Kleinschmidt (Max Planck--AEI)
Title: Fourier coefficients and nilpotent orbits for small representations
Abstract: Small representations are distinguished by the fact that they have very few non-trivial Fourier coefficients and there are known connections between wavefront sets and nilpotent orbits. These concepts can also be connected to properties of degenerate Whittaker vectors which are fully computable even in the Kac-Moody case.

Speaker: Kyu-Hwan Lee (UConn)
Title: Convergence of  Kac-Moody Eisenstein Series
Abstract: We consider  Eisenstein series on an arbitrary Kac-Moody  group,  induced from quasi-characters, and prove the almost-everywhere convergence of  Kac-Moody Eisenstein series inside the Tits cone for spectral parameters in the Godement range.  For a certain class of Kac-Moody groups satisfying an additional combinatorial property, we show absolute convergence everywhere in the Tits cone for spectral parameters in the Godement range. This is achieved by establishing the absolute convergence of the constant terms and showing that the Eisenstein series are essentially dominated by the constant terms. This is a joint work with Carbone, Garland, Liu and Miller.

Speaker: Dongwen Liu (Zhejiang)
Title: Rank 2 Kac-Moody Eisenstein series over finite fields
Abstract: For a rank 2 Kac-Moody group G over a finite field, we define Eisenstein series on the quotient graph of the Tits building X by a negative parabolic subgroup using eigenfunctions of adjacency operator. We establish convergence of Eisenstein series in a half space by proving an Iwasawa decomposition of the Haar measure on G. We prove meromorphic continuation of Eisenstein series using the Selberg-Bernstein continuation principle. This requires an analog of the classical truncation operator for Eisenstein series and integral operators on the Tits building whose convolution kernels are bounded radial functions on X. This is a joint work with L. Carbone, H. Garland and D. Gourevich.

Speaker: Sergey Oblezin (Nottingham)
Title: Whittaker functions and Harmonic analysis
Abstract: Whittaker functions were introduced by Jacquet and Langlands in the end of 1960s, and since then play an important role in Harmonic Analysis for non-compact reductive groups. The non-Archimedean local Langlands correspondence provides a relation between the Whittaker function over a local field and complete flag manifold of the complex dual group.
In 2009 jointly with A.Gerasimov and D.Lebedev, a new representation of Whittaker functions and automorphic L-functions for GL(N,R) in the framework of 2-dimensional topological field theories was proposed. In this setting the Archimedean Langlands correspondence is identified with the Mirror Symmetry of the underlying TFTs. In my talk I'll outline the main constructions of this approach in the framework of Harmonic Analysis. In the end of my talk I'll present the applications including the Archimedean and the q-defomation analogs of the Langlands-Shintani formula, and discuss further directions.

Speaker: Manish Patnaik (Alberta)
Title: Cuspidal Eisenstein Series over Function Fields
Abstract: I will explain the construction of cuspidal Eisenstein series on loop groups over function fields starting from both ‘positive’ and ‘negative’ maximal parabolics. A functional equation can be proven that relates the positive and negative cuspidal Eisenstein series and in which L-functions of finite-dimensional cusp forms appear. This is based on works with: A. Braverman, H. Garland, D. Kazhdan, and S.D. Miller.

Speaker: Daniel Persson (Chalmers)
Title: Degenerate Whittaker vectors and small automorphic representations of exceptional groups
Abstract: I will demonstrate how certain Fourier coefficients of automorphic forms attached to the minimal representations of E6, E7, E8 are determined by maximally degenerate Whittaker vectors. This fact allows for a simple method for calculating explicit Fourier coefficients which are relevant in string theory.

Speaker: Guy Rousseau (Université de Lorraine)
Title: Kac-Moody groups over local fields: building hovels
Abstract: In this talk, I explain the construction of the affine hovels associated to Kac-Moody groups over local fields. They are the analogue of the Bruhat-Tits buildings associated to reductive groups. I shall explain the similarities and differences with these buildings.

Speaker: Ian Whitehead (Minnesota)
Title: Generalizations of the Casselman-Shalika Formula
Abstract: The Casselman-Shalika formula characterizes the Whittaker function for unramified principal series representations of p-adic groups. I will describe extensions of this formula to metaplectic groups, affine Kac-Moody groups, and beyond.

Speaker: Chenyan Wu (Fudan)
Title: Theta Correspondence, Certain Periods and Arthur Parameters
Abstract: Let $sigma$ be a cuspidal automorphic representation of a unitary group. We show a three-way relation among first occurrence of $sigma$ in the tower of theta correspondence of unitary groups, certain periods of $sigma$ and certain periods of residue of Eisenstein series associated to $sigma$. This gives a way to detect certain simple factors of the form $(chi, b)$ in the Arthur parameter of $sigma$ where $chi$ is a character and $b$ is an integer. Analogous results for other dual reductive pairs should also hold.

Speaker: Yichao Zhang (Harbin)
Title: Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras
Abstract: In the first part of this talk, we shall introduce reduced weakly holomorphic modular forms with sign vectors. Such modular forms correspond to some vector-valued modular forms of full level and we shall see some of their "full-level" properties. By passing to vector-valued modular forms, the Borcherds lifts of such modular forms provide us natural candidates for automorphic corrections of rank two hyperbolic Kac-Moody algebras. In the second part, as a joint work with Henry H. Kim and Kyu-Hwan Lee, we shall see three more examples of such autmorphic correction, extending a previous work of Henry H. Kim and Kyu-Hwan Lee.

Speaker: Yongchang Zhu (HKUST)
Title: Theta Lifting of Automorphic Forms for Loop Groups
Abstract: In this talk, I discuss the theta lifting of automorphic forms on finite dimensional groups to the loop groups.