Akihiko Yukie
In this paper, we give an introduction to the rationality of the equivariant Morse stratification, and state the author's results on zeta functions of prehomogeneous vector spaces.
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This is part one of a series of papers. In this series of papers, we consider problems analogous to the Oppenheim conjecture from the viewpoint of prehomogeneous vector spaces.
In this paper, we construct a new series of prehomogeneous vector spaces from figures made up of triangles, called triangle arrangements. Our main theorem states that, under suitable assumptions, we are able to construct a prehomogeneous vector space obtained from a triangle arrangement by attaching two triangle arrangements corresponding to prehomogeneous vector spaces at a vertex. We also give examples of prehomogeneous vector spaces obtained from triangle arrangements. Many of them seem to be new.
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 a 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 …
We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.
A functional equation between the zeta distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.
In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced in [CFGK17]. The second identity is a generalization of the descent construction studied in [GRS11].
We consider the residues at the poles in the right half plane of Eisenstein series, on symplectic groups, or their double covers, induced from Speh representations. We show that for each such pole, there is a unique maximal nilpotent orbit, attached to Fourier coefficients admitted by the corresponding residual representation. We find this orbit in each case.
Whittaker coefficients of geometric Eisenstein series
本文通过将几何 Eisenstein 级数的 Whittaker 系数解释为因子分解同调，并利用 Beilinson 和 Drinfeld 的手性同调公式，证明了几何 Langlands 预言中 Whittaker 系数与局部系统模空间函数间的同构关系。
Based on Garland's work, in this paper we construct the Eisenstein series on the adelic loop groups over a number field, induced from either a cusp form or a quasi-character which is assumed to be unramified. We compute the constant terms, prove their absolute and uniform convergence under the affine analog of Godement's criterion. For the case of quasi-characters the resulting formula is an affine Gindikin-Karpelevich formula. Then we prove the convergence of Eisenstein series themselves in certain analogs of Siegel subsets.
This paper introduces the template method for computing the first coefficient of Langlands Eisenstein series on $\GL(n,\mathbb R)$ and more generally on Chevalley groups over the adele ring of $\mathbb Q.$ In brief, the first coefficient of Borel Eisenstein series can be used as a template to compute the first coefficient of more general Eisenstein series by elementary linear algebra calculations.
Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.
We review some topics in the analytic theory of Eisenstein series, including meromorphic continuation, $L^2$-spectral expansion and Fourier coefficients. We also discuss some open problems.
In this paper we obtain a complete description of images and poles of degenerate Eisenstein series attached to maximal parabolic subgroups of $Sp_4(\mathbb A)$, where $\mathbb A$ is the ring of adeles of $\mathbb Q$.
I this paper we determine poles in the right--half plane and images of degenerate Eisenstein series for $GL_n(\Bbb A_\mathbb Q)$ induced from a character on a maximal parabolic subgroup. We apply those results to determine poles of degenerate Eisenstein series for $GL_n(\Bbb R)$.
Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove that the cuspidal Eisenstein series are entire on the full complex plane.
We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculation gives the Godement-Jacquet zeta function. This realizes and generalizes the construction proposed by Ginzburg and Soud…
Eisenstein Series on Metaplectic Covers and Multiple Dirichlet Series
本文计算了整体元辛群上由环面诱导的爱森斯坦级数的首个惠特克系数，并将其与对偶根系统的外尔群多重狄利克雷级数建立联系，从而证实了 Brubaker-Bump-Friedberg 猜想。
We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.
We set up a general framework to compute the exact multiplicity with which certain automorphic representations appear in both the cuspidal and Eisenstein cohomology of locally symmetric spaces. We apply this machinery to Eisenstein series on $GSp_4$ and split $G_2$. In the case of $G_2$, we will also obtain new information about the archimedean components of certain CAP representations using Arthur's conjectures.
We have shown in a preceeding paper how to normalize intertwining operators for classical groups using the twisted endoscopy lifting. In this paper, we prove that the image of such an operator in the cases interesting in the theory of Eisenstein Series, is either 0 or an irreducible representation. As a consequence we compute explicitly the points where Eisenstein Series for square integrable representations are not holomorphic under some hypothesis at the archimedean places: at that places we mainly assume that the infinitesimal character is integral and regular.
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, who previously showed the analogous Eisenstein series are finite sums.
Fourier Coefficients of the Degenerate Eisenstein Series on Symplectic Groups
该研究针对辛群上的退化艾森斯坦级数构造新自守下降，证明特定傅里叶系数限制后仍为同参数艾森斯坦级数，明确全纯截面形式，证明局部正则点的相关性质，并指出迭代下降的应用。
In our earlier paper, based on a paper by Bump and Ginzburg, we used an Eisenstein series on the double cover of GL(r) to obtain an integral representation of the twisted symmetric square L-function of GL(r). Using that, we showed that the (incomplete) twisted symmetric square L-function of GL(r) is holomorphic for Re(s) > 1. In this paper, we will determine the possible poles of this Eisenstein series more precisely and show that the (incomplete) twisted symmetric square L-function is entire except possible simple poles at s = 0 and s = 1.
The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic, whose existence is non-obvious from the point of view of classical tools.
Parabolic geometric Eisenstein series and constant term functors
该研究证明惠特克层抛物限制与几何卡塞尔曼-沙利卡等价下表示限制的相容性，建立几何艾森斯坦级数函子的赫克结构并推广相关结果，还通过Koszul对偶联系两类艾森斯坦级数函子，概述几何朗兰兹函子与常数项函子可交换的证明。
Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_9(R)$, $E_{10}(R)$ and $E_{11}(R)$ corresponding to certain degenerate principal series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings $R^4$ and $\partial^{4} R^4$ coming from 1/2-BPS and 1/4-BP…
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