Eisenstein Series on Affine Kac-moody Groups over Function Fields

In his pioneering work, H. Garland constructed Eisenstein series on affine Kac-Moody groups over the field of real numbers. He established the almost everywhere convergence of these series, obtained a formula for their constant terms, and proved a functional equation for the constant terms. In his subsequent paper, the convergence of the Eisenstein series was obtained. In this paper, we define Eisenstein series on affine Kac-Moody groups over global function fields using an adelic approach. In the course of proving the convergence of these Eisenstein series, we also calculate a formula for the constant terms and prove their convergence and functional equations. Introduction Classical Eisenstein series are central objects in the study of automorphic forms. The classical Eisenstein series were generalized to the case of reductive groups and studied by R. Langlands [13,14]. As in the classical case, he found these Eisenstein series have certain analytic properties as well as Fourier series expansions where L-functions appear in the constant terms. Because of this relationship, these Lfunctions inherit important analytic properties from the Eisenstein series. This approach to studying automorphic L-functions is known as the Langlands-Shahidi method. (See [7] for a survey.) The Eisenstein series over function fields were studied by G. Harder in [8]. In [4], H. Garland defines and studies Eisenstein series on affine Kac-Moody group over R. He proved the almost everywhere convergence of the series while placing specific emphasis on calculating the constant term, finding its region of convergence, and proving functional equations of the constant term. More precisely, let ĜR be an affine Kac-Moody group over R. For each character χ of the positive part  of the torus, he defines a function Φχ :  → C× and, as in the classical case, uses the Iwasawa decomposition ĜR = K̂ Â Û to extend Φχ to a function on ĜR. Garland extends this group by the automorphism e −rD ∈ Aut(V λ R ), where r > 0 and D is the degree operator of the Kac-Moody Lie algebra associated with ĜR. Setting Φχ(ge −rD) = Φχ(g), he defines an Eisenstein series Eχ on the space ĜR e −rD ⊆ Aut(V λ R ) by ∑ γ∈Γ̂/(Γ̂∩B̂) Φχ(ge −rDγ), where Γ̂ is a discrete subgroup of ĜR. Received by the editors February 8, 2011 and, in revised form, August 15, 2012. 2010 Mathematics Subject Classification. Primary 22E67; Secondary 11F70. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication