Top Fourier coefficients of residual Eisenstein series on symplectic or metaplectic groups induced from Speh representations

Whittaker Coefficients of Metaplectic Eisenstein Series

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For "cominuscule" maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing …

Metaplectic Ice for Cartan Type C

We use techniques from statistical mechanics to provide new formulas for Whittaker coefficients of metaplectic Eisenstein series on odd orthogonal groups, matching Friedberg and Zhang. We study a particular variation/generalization of the six-vertex model of Cartan type C having "domain-wall boundary conditions" dependent on a given integer partition $λ$ of length at most $r$, where $r$ is a fixed positive integer. More precisely, we examine a planar, non-nested, U-turn model whose partition functions $Z_λ$ are a generalization of a deformation of characters of the symplectic group $\operatorn…

$GL(n,\mathbb{A}_\mathbb{Q})$的Eisenstein级数极点相消中的一些组合学

Some Combinatorics in the Cancellation of Poles of Eisenstein Series for $GL(n,\mathbb{A}_\mathbb{Q})$

本文通过描述相关Weyl群陪集的组合学，将退化Eisenstein级数的极点相消方法与结果推广至由两个Speh表示诱导构造的Eisenstein级数。

扭曲雅克比模与从斯佩表示的诱导

Twisted Jacquet modules and induction from Speh representations

本文研究非阿基米德局部域上的扭曲雅克比模，证明其与斯佩表示诱导的相关性质，确定例外参数集，并给出辛二重覆盖及大特征有限域上的对应结果。

A Godement-Jacquet type integral and the metaplectic Shalika model

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

Geometric Whittaker models and Eisenstein series for Mp_2

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Let Bun_{Mp_2} be the stack of metaplectic bundles on X of rank 2. In this paper we study the derived category of genuine l-adic sheaves on Bun_{Mp_2} in the framework of the quantum geometric Langlands. We describe the corresponding Whittaker category, develop the theory of geometric Eisenstein series and calculate the most non-degenerate Fourier coefficients of these Eisenstein series. The existing constructions of automorphic sheaves for GL_n are based on using Whittaker sheaves. Our calculations lea…

Notes on Analytic Properties of Residual Eisenstein Series, I

We partially generalize the results of Kudla and Rallis on the poles of degenerate, Siegel-parabolic Eisenstein series to residual-data Eisenstein series. In particular, for $a,b$ integers greater than 1, we show that poles of the Eisenstein series induced from the Speh representation $Δ(τ,b)$ on the Levi $\mathrm{GL}_{ab}$ of $\mathrm{Sp}_{2ab}$ are located in the "segment" of half integers $X_{b}$ between a "right endpoint" and its negative, inclusive of endpoints. The right endpoint is $\pm b/2$, or $(b-1)/2$, depending on the analytic properties of the automorphic $L$-functions attached to…

Doubling Constructions and Tensor Product ${L}$-Functions: the linear case

We present an integral representation for the tensor product $L$-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. This is the first in a series of papers, treating…

The double cover of odd general spin groups, small representations and applications

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series, it is an automorphic representation and decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of SO(2n+1) of Bump, Fr…

Criteria for the Existence of Cuspidal Theta Representations

Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of $GL_j$, $j\leq 3$. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the $r$-fold metaplectic cover of the general linear group of arbitrary rank.

Weyl Group Multiple Dirichlet Series for Symmetrizable Kac-Moody Root Systems

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells' work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorph…

Weyl group multiple Dirichlet series of type C

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. In type C, they conjecturally arise from the Fourier-Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold metaplectic cover of SO(2r+1). For any odd n, we construct an infinite family of Dirichlet series conjecturally satisfying the above analytic properties. The coefficients of …

Fourier-Jacobi周期与$\boldsymbol{\text{Mp}}_{2n} \times \text{Sp}_{2m}$的非缓增Gan–Gross–Prasad猜想

Fourier-Jacobi periods and the non-tempered Gan--Gross--Prasad conjecture for $\Mp_{2n} \times \Sp_{2m}$

本文针对两类非缓增全局$A$-参数族，分别用正则化Fourier-Jacobi周期与全局theta对应等方法，建立了辛-亚辛对的非缓增全局GGP猜想的一个方向，还得到了缓增GGP猜想的相关结果与$L$-包二分性结论。