Entirety of cuspidal Eisenstein series on loop groups

Abstract

Abstract. In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the finite-dimensional setting, where such series only converge absolutely in a right half plane (and have poles elsewhere coming from L-functions in their constant terms). Our result is the Q-analog of a theorem of A. Braverman and D. Kazhdan from the function field setting [4, Theorem 5.2], who previously showed the analogous Eisenstein series there are finite sums.

Secondary terms in the asymptotics of moments of L-functions

“…Let all notations be as before. By using (20), (21) and Proposition 5.1, it is not hard to check that lim…”

“…Presumably, this infinite product is structurally similar to the denominator of the character of an irreducible highest-weight g(A)-module, which converges on the interior of the complexified Tits cone of W. Thus the constant term would be meromorphic on X * 0 , and by a well-known principle, the minimal parabolic Eisenstein series should have meromorphic continuation to the same domain. However, developing the Langlands-Shahidi method in the context of Kac-Moody groups is quite problematic at the moment, a serious obstacle in doing so being the lack of an adequate integration theory over the relevant unipotent radicals allowing us to transfer the "meromorphy property" from an Eisenstein series to its Whittaker coefficients; see [20] and the reference therein for specific information in the special case of Eisenstein series on loop groups. Nonetheless, in some special cases, it is possible to obtain the meromorphic continuation of the corresponding Weyl group multiple Dirichlet series more directly (see, for example, [8], [40], and especially the forthcoming manuscripts [14], [15], where the conjecture is proved in the important case of untwisted Weyl group multiple Dirichlet series of type D (1) 4 over rational function fields).…”

Adelic descent theory

“…We refer the reader to § 3.2.4 for a more detailed discussion of the adelic description of -bundles on Noetherian schemes . The case of punctured surfaces has also been considered by Garland and Patnaik in [GP]. In [Par83], Parshin used adelic cocycles for -bundles as above to obtain formulae for Chern classes in adelic terms.…”

Theta lifting for loop groups

“…Subsequent works in this area include studies of representations of loop group over p-adic fields through various sophisticated techniques [BK1,BGKP,BKP,P2], which will have important applications to automorphic forms. See also [P1,Li1,GMM,LL] for the following works on loop Eisensteins series, and [CLL] for an example of Kac-Moody Eisenstein series beyond the loop case. We refer the readers to [BK2] for a more complete list of literatures.…”