The Siegel-Weil formula for unitary groups: the second term range

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

We formulate and prove the Siegel-Weil formula for loop groups.

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。

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

We proved a new Siegel-Weil formula for orthogonal and symplectic groups, which will be used later to prove a generalization of Siegel-Weil formula for loop groups.

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。

A new regularized Siegel-Weil type formula, part I

In this paper, we propose a formula relating certain residues of Eisenstein series on symplectic groups. These Eisenstein series are attached to parabolic data coming from Speh representations. The proposed formula bears a strong similarity to the regularized Siegel-Weil formula, established by Kudla and Rallis for symplectic-orthogonal dual pairs. Their work was later generalized by Ikeda, Moeglin, Ichino, Yamana, Gan-Qiu-Takeda and others.

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。

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…

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。

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。

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。

有限域上酉群 Weil 表示的 Shintani 提升

Shintani lifts for Weil representations of unitary groups over finite fields

该研究通过几何构造给出有限域上酉群的扩展 Weil 表示，并证明其构成相应 Weil 表示的 Shintani 提升。

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.

有限酉群的模$\boldsymbol{\text{ell}}$Weil表示与Deligne--Lusztig诱导

Mod $\ell$ Weil representations and Deligne--Lusztig inductions for unitary groups

该研究聚焦有限酉群的模$\text{ell}$Weil表示与相关Deligne--Lusztig诱导，分析其作为辛群表示的分解，构建了含$p=2$情形的$(\text{Sp}_{2n},\text{O}_2^-)$模$\text{ell}$Howe对应。

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…

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodro…

有限酉群的 Heisenberg-Weil 表示的几何构造与 Howe 对应

Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

本文通过代数簇的中阶 étale 上同调几何构造了有限酉群的 Heisenberg-Weil 表示，并利用 Frobenius 作用实现了任意有限域上 $(\mathit{Sp}_{2n},O_2^-)$ 的 Howe 对应，证明了幂单性在该对应下保持不变。

酉群Weil表示的环面周期

On tori periods of Weil representations of unitary groups

本文确定了酉群Weil表示到极大环面的限制，证明了局部相容特征与全局环面周期非零的充要条件，并利用跷跷板论证和theta提升结果为构造G2的Arthur包提供了基础。

Weil representations of unitary groups over ramified extensions of finite local rings with odd nilpotency length

We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of the maximal ideal of $A$ is odd. We show in particular that this Weil representation is multiplicity free. Restriction to the special unitary group $\mathrm{SU}_{2n}(A)$ preserves irreducibility and multiplicity freeness provided $n>1$.

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.

Clifford theory of Weil representations of unitary groups

Let ${\mathcal O}$ be an involutive discrete valuation ring with residue field of characteristic not 2. Let $A$ be a quotient of ${\mathcal O}$ by a nonzero power of its maximal ideal and let $*$ be the involution that $A$ inherits from ${\mathcal O}$. We consider various unitary groups ${\mathcal U}_m(A)$ of rank $m$ over $A$, depending on the nature of $*$ and the equivalence type of the underlying hermitian or skew hermitian form. Each group ${\mathcal U}_m(A)$ gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil…

Weil representation and Arithmetic Fundamental Lemma

By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $\mathbb{Q}_p$ when $p\geq n$.

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…

The Weil-Steinberg character of finite classical groups

We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an…

On the geometry of Siegel-Jacobi domains

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.

无重数表示与拟对称Siegel域上某些幂零Lie群的余迷向作用

Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains

本文研究拟对称Siegel域上两步幂零Lie群的无重数表示，给出了该性质成立的充要条件，并证明了不可约酉表示的不相交性、Bergman空间上酉表示的无重数性与群作用余迷向性的等价关系。

Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$, $q>3$ odd via presentation and compatibility of methods

In this article we construct Weil representations of quasi-split unitary groups $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associated to quadratic extensions of finite fields. We define these representations by using an adequate presentation Bruhat like of those groups. More precisely, we define Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associating to each generator a linear map of a suitable $\mathbb{C}$-vector space satisfying the relations of the aforementioned presentation. In addition, we also address the natural question on the compatibility of our representation of $U…

Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

We construct a complex linear Weil representation $ρ$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $ρ…

Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms

Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $Π_{i=1}^{r} SL_{n_i} \subseteq M \subseteq Π_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under \textit{the local Jacquet-Langlands type correspondence} between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Mui{ć} and Savin (2000)…

Weil representations via abstract data and Heisenberg groups: a comparison

Let $B$ be a ring, not necessarily commutative, having an involution $*$ and let ${\mathrm U}_{2m}(B)$ be the unitary group of rank $2m$ associated to a hermitian or skew hermitian form relative to $*$. When $B$ is finite, we construct a Weil representation of ${\mathrm U}_{2m}(B)$ via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of ${\mathrm U}_{2m}(B)$, and we compare the two Weil representations thus obtained under …

Quaternion distinguished generic representations of $\mathrm{GL}_{2n}$

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representa…

Archimedean zeta integrals on U(2,1)

It is known that for a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions, which have applications to arithmetic. In this paper we explicitly calculate archimedean zeta integrals of this type for U(2,1). In particular we compute the matrix coefficients of Weil representations using joint harmonics in the Fock model, and those of discrete series using Schmid operators.

Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existenc…

一个p进算术内积公式

A $p$-adic arithmetic inner product formula

本文构造了偶秩拟分裂酉群上自守表示的分圆p进L-函数，证明其在平凡特征处消没阶为1时Selmer群秩至少为1，并利用酉Shimura簇上的特殊圈构造Selmer theta提升，建立了p进高度与p进L-函数导数之间的精确公式，即算术内积公式的p进类比。

对称对$(U(p,p),GL_{p}(\boldsymbol{\text{C}}))$的实双旗簇与伽罗瓦上同调

Real double flag variety for the symmetric pair $(U(p,p),GL_{p}(\mathbb{C}))$ and Galois cohomology

本文针对对称对$(U(p,p),GL_{p}(\text{C}))$，在$p=2$时利用伽罗瓦上同调对实双旗簇的轨道分解$H\ackslash (H/B_H\ imes G/P_S)$完成了分类。