On the Siegel-Weil Theorem for Loop Groups (I)

An exceptional Siegel-Weil formula and poles of the Spin L-function of $PGSp_6$

We show a Siegel-Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin-Selberg integral for the Spin L-function of $PGSp_6$ discovered by A. Pollack, we prove that a cuspidal representation of $PGSp_6$ is a (weak) functorial lift from the exceptional group $G_2$ if its (partial) Spin L-function has a pole at $s=1$.

模曲线乘积上的算术Siegel–Weil公式

Arithmetic Siegel--Weil formula on the product of modular curves

该研究证明了两条模曲线乘积上带任意奇异系数的完整生成级数算术Siegel–Weil公式，对应Kudla纲领的GSpin(2,2)情形。

A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Following W.T.Gan and S.Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$, where $V$ is a split hermitian space of dimension $2r$ with $r+1 \leq n \leq 2r-1$. As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$.

余秩 $1$ 算术 Siegel–Weil IV：解析局部到整体

Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global

本文作为系列论文的第四篇，通过确定 Siegel Eisenstein 级数归一化、建立局部 Siegel–Weil 特殊值公式并拼接局部主要定理，完成了余秩 $1$ 算术 Siegel–Weil 公式的证明。

余秩 $1$ 算术 Siegel–Weil II：局部阿基米德情形

Co-rank $1$ Arithmetic Siegel--Weil II: Local Archimedean

本文作为系列论文第二篇，证明余秩1情形下关键阿基米德局部定理及纯阿基米德相交数下的局部算术Siegel–Weil公式，通过新极限方法将Green流与局部Whittaker函数离中心导数相联系。

余秩 $1$ 算术 Siegel–Weil III：几何局部到整体

Co-rank $1$ Arithmetic Siegel--Weil III: Geometric local-to-global

本文作为系列论文第三篇，完成余秩1算术Siegel–Weil公式中从整体算术交数到局部几何量的约化，并提出任意余秩奇异矩阵的算术特殊循环类构造。

与Weil表示相关的Siegel模形式

Siegel modular forms associated to Weil representations

该研究利用2-上同调分析与Weil表示相关的Siegel模形式的八次单位根，将研究拓展至完全Siegel群，得到两个矩阵值Siegel模形式并明确其对应的有限维表示。

余秩 $1$ 算术 Siegel–Weil I：局部非阿基米德情形

Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean

本文作为系列论文首篇，提出新的局部极限方法，证明所有非阿基米德位的关键局部定理，为证明余秩1算术Siegel–Weil公式奠定基础。

On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let $π$ be a cuspidal automorphic representation of an orthogonal group $O(V)$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that there is a place $v$ such that $π_v\congπ_v\otimes\det$. Then the global theta lift of $π$ to $Sp_{2r}$ does not vanish up to twisting by automorphic dete…

酉群的高阶 Siegel–Weil 公式：非奇异项

Higher Siegel--Weil formula for unitary groups: the non-singular terms

本文在酉 shtuka 模叠上构造特殊闭链，证明归一化 Siegel–Eisenstein 级数非奇异 Fourier 系数的 r 阶中心导数等于虚拟维数 0 特殊闭链的度数，从而给出 Kudla–Rapoport 猜想的函数域类比并涵盖所有高阶导数。

分裂例外型 $E_n$ 群上退化爱森斯坦级数的极点、留数与 Siegel-Weil 恒等式

Poles, Residues and Siegel-Weil Identities of Degenerate Eisenstein Series on Split Exceptional Groups of Type $E_n$

本文系统研究分裂例外型 $E_n$（$n=6,7,8$）群的退化剩余谱，完整描述 $E_6$、$E_7$ 情形并几乎完整描述 $E_8$ 情形，同时给出 Siegel-Weil 型恒等式与 Arthur 参数。

紧 $\mathrm{Spin}_8$ 的例外 Siegel–Weil 定理

Exceptional Siegel-Weil theorems for compact $\mathrm{Spin}_8$

本文研究四元数 $E_8$ 中 $G_E\times S_E$ 对偶对，利用极小表示计算平凡表示的提升，证明其对应退化 Eisenstein 级数，并用于 $G_2$ 模形式算术性。

The Schroedinger-Weil Representation and Jacobi Forms of Half-Integral Weight

In this paper, we define the concept of Jacobi forms of half-integral weight using Takase's automorohic factor of weight 1/2 for a two-fold covering group of the symplectic group on the Siegel upper half plane and find covariant maps for the Schroedinger-Weil representation. Using these covariant maps, we construct Jacobi forms of half-integral weight with respect to an arithmetic subgroup of the Jacobi group.

西格尔zeta函数的模性

The modularity of Siegel's zeta functions

本文利用新近的Maass形式Weil型逆定理证明了西格尔不定二次型zeta函数的模性，完成了西格尔1938年提出的研究设想，同时证明这类zeta函数的一半对应全纯模形式。

Generating series of intersection volumes of special cycles on unitary Shimura varieties

We form a generating series of regularized volumes of intersections of special cycles on a non-compact unitary Shimura variety with a fixed base change cycle. We show that it is a Hilbert modular form by identifying it with a theta integral, which we show converges even though the parameters lie outside the classical convergence range of Weil. By applying the regularized Siegel-Weil formulas of Ichino and Gan-Qiu-Takeda, we show the modular form is the restriction of a hermitian modular form of degree n related to Siegel Eisenstein series on U(n,n). An essential fact used is a computation show…

Weil representations of twisted loop groups of type $A_n^{(2)}$

We construct Weil representations of twisted loop groups of type $A_n^{(2)}$ over local fields. We prove that the associated cover of the twisted loop group is the two fold metaplectic cover of the affine Kac-Moody group of type $A_n^{(2)}$ given by Patnaik-Puskas.

Boundary behavior of special cohomology classes arising from the Weil representation

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the t…

Eisenstein Series on Loop Groups

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.

Degenerate principal series of metaplectic groups and Howe correspondence

The main purpose of this article is to supplement the authors' results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these representations play a key role in understanding the basic structure of Howe correspondence (and its complications in the archimedean case), and their global counterparts play an equally essential part in the…

Discrete Series for Loop Groups.I. An algebraic Realization of Standard Modules

In this paper we consider the category $C (\tilde k, \tilde H)$ of the $(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$ are the corresponding affine Lie algebra and the affine Cartan group, respectively. To this category we apply the Zuckerman functor and its derivatives. By using the determinant bundle structure, we prove the natural duality of the Zuckerman derived functors, and deduce a Borel-Weil-Bott type theorem on decomposition of the nilpotent part cohomology.