Dongwen Liu, Zhicheng Wang
In this paper we study the Gan-Gross-Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.
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In this paper we study the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.
The Gan-Gross-Prasad problem is to describe the restriction of representations of a classical group $G$ to smaller groups $H$ of the same kind. In this paper, we solved the Gan-Gross-Prasad problem over finite fields completely. In previous work \cite{LW1,LW2,LW3,Wang}, we study the Gan-Gross-Prasad problem for unipotent representations of finite classical groups. The main tools used are the Lusztig correspondence as well as a formula of Reeder \cite{R} for the pairings of Deligne-Lusztig characters. We give a reduction decomposition of Reeder's formula, and deduce the Gan-Gross-Prasad problem…
The twisted Gan-Gross-Prasad problem for finite classical groups
本文研究有限域上经典群的扭曲Gan-Gross-Prasad问题，推导了Deligne-Lusztig特征的乘法公式，并给出了来自椭圆tori的不可约表示的完整解答。
In this paper, we give a proof of the Gan-Gross-Prasad conjecture for the discrete series of U(p,q). Given a discrete series representation $D(λ)$ in terms of the Harish-Chandra parameter, the restriction of $D(λ)$ to U(p-1,q) contains $D(μ)$ as a subrepresentation if and only if $λ$ and $μ$ interlaces in a very special way.
Ext-analogues of Gan--Gross--Prasad models vanish for tempered representations
本文证明了 Gan–Gross–Prasad 模型的 Ext 类比在温和表示下为零，从而支持 D. Prasad 的猜想，并由此得到猜想性 Euler–Poincaré 特征公式，同时证明包含 Steinberg 表示在内的一类特殊温和表示的限制是投射的。
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups II: Comparison of the relative trace formulae
本文通过比较相对迹公式，证明了单位ary群上傅里叶-雅可比周期的广义Gan-Gross-Prasad猜想及其改进。
We prove the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case under some assumptions. This is the last case of the local Gan-Gross-Prasad conjectures. We also prove two of Prasad's conjectures on the local theta correspondence in the almost equal rank case.
We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.
We prove the local Gross-Prasad conjecture for generic L-packets of representations of special orthogonal groups. The proof uses the same result for tempered L-packets proved in a preceding paper, and irreducibility results for the induced representations of whose the elements of the L-packets are Langlands quotients.
Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan-Gross-Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.
Spherical Characters in Families: the unitary Gan-Gross-Prasad case
本文研究了族中球面特征的变分，提出了其有理性与亚纯性质的猜想，并在酉 Gan-Gross-Prasad 情形下证明了这些猜想。
Twisted Gan-Gross-Prasad conjecture for unramified quadratic extensions
本文采用相对迹公式，在非分歧性与局部条件下证明了酉群嵌入一般线性群情形的扭曲全局 Gan-Gross-Prasad 猜想及其精细化。
The Local Gross-Prasad Conjecture over Archimedean Local Fields
该研究采用C. Moeglin和J.L.-Waldspurger的方法，完成了阿基米德局部域上局部Gross-Prasad猜想一般情形的证明，推进了相关数论问题的研究。
The local twisted Gan-Gross-Prasad conjecture for $\text{U}(V_K)/\text{U}(V)$
本文通过局部迹公式及其扭曲变体的几何展开，证明了在奇剩余特征非阿基米德域上对于缓增$L$-参数的$U(V_K)/U(V)$局部扭曲Gan-Gross-Prasad猜想。
We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our mains theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups.
We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n)\times\mathrm{U}(n+1)$ in the stable case.
We formulate a global Gan-Gross-Prasad conjecture for general spin groups. That is, we formulate a conjecture on a relation between periods of certain automorphic forms on $GSpin_{n+1} \times GSpin_n$ along the diagonal subgroup $GSpin_n$ and some $L$-values. To support the conjecture, we show that the conjecture holds for $n=2$ and $3$ and for certain cases for $n=4$.
Through combining the work of Jean-Loup Waldspurger (\cite{waldspurger10} and \cite{waldspurgertemperedggp}) and Raphaël Beuzart-Plessis (\cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (\cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.
In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(π)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.
We consider irreducible unitary representations $A_i$ of G=SO(n+1,1) with the same infinitesimal character as the trivial representation and representations $B_j$ of H=SO(n,1) with the same properties and discuss H-equivariant homomorphisms Hom_H($A_i,B_j$). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.
We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.
We prove the local Gross-Prasad conjecture for tempered representations of special orthogonal groups. Roughly speaking, the conjecture says that, if sigma is an irreducible representation of SO(n) and rho is an irreducible representation of SO(n-1), rho appears as quotient of the restriction of sigma to SO(n-1) with a multiplicity m(sigma,rho) that can be computed in terms of epsilon-factors. Our proof uses results of a previous papers which computes m(sigma,rho) and the epsilon-factors by integral formulas.
Let $F$ be a non-archimedean local field, of characteristic 0. Let $V$ be a finite dimensional vector space over $F$ and $q$ be a non-degenerate quadratic form on $V$. Denote $G$ the special orthogonal group of $(V,q)$. Let $W$ a non-degenerate hyperplane of $V$, denote $H$ the special orthogonal group of $W$. Let $π$, resp. $σ$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(σ,π)$ the dimension of the complex space $Hom_{H(F)}(π_{| H(F)},σ)$. It's know that $m(σ,π)=0$ or 1. In a first paper, we have defined another term $m_{geom}(σ,π)$. It's an explicit sum of int…
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae
本文首次证明了单位群上Fourier-Jacobi周期的全局Gan-Gross-Prasad猜想，并建立了相对迹公式的大规模展开。
The local Gross-Prasad conjecture over $\mathbb{R}$: Epsilon dichotomy
本文基于Waldspurger的工作，证明了实数域上 tempered 局部 L-参数的局部格罗斯-普拉兹猜想的 ε 二分法部分。
Quotient branching laws and unitary Gan-Gross-Prasad relevance for general linear groups
本文研究一般线性群中不可约光滑表示的分支定律，提出算法确定Gan-Gross-Prasad相关对，并对广义Speh表示进行分类。
Regularized Periods and the global Gan-Gross-Prasad conjecture : The case of $U(n+2r) \times U(n)$
该研究在非约化群上引入正则化三线性周期，既可为典型群、亚辛群定义两类正则化周期，又能证明斜埃尔米特酉群上Gan-Gross-Prasad猜想的一个方向。
The local twisted Gan-Gross-Prasad conjecture for $\text{GL(V)/U(V)}$
本文利用 Waldspurger 与 Beuzart-Plessis 的 Bessel 模型方法研究两个局部相对迹公式，通过谱展开与部分几何比较，在非阿基米德域上对 tempered 表示证明了 GL(V)/U(V) 的局部扭曲 Gan-Gross-Prasad 猜想。
We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson product against the product of two elliptic cuspidal Hecke eigenforms. The square of this integral can be explicitly expressed in terms of the central critical value of an L-function attached to the situation. The result is related to a conjecture of Gross and Prasad about restrictions of automorphic representations of special orthogonal groups.
We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F.
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems
本文证明了关于单位ary群上的Fourier-Jacobi周期的Gan-Gross-Prasad猜想，并通过谱展开和零阶情况的减少完成了主要定理的证明。
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic 0 and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitian spaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonal complement of $W$ is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over $F$). Let $π$ and $σ$ be smooth irreducible representations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasad have defined a multiplicity $m(π,σ)$ which for $m=n-1$ is just the dimension of $Hom_{H(F)}(π,σ)$. For $π$ and $σ$ tempered, we state and prove an integral formula f…
Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures
本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。
Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture
本文在相对 Langlands 纲领下研究了涉及 Weil 表示的新相对迹公式，通过几何分解与全局比较，在局部假设下证明了任意维度且允许数域分歧的扭曲 GGP 猜想。
On the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$
本文一般地证明了酉群对 (U(2n),U(1)) 的 Gan-Gross-Prasad 猜想，并在一定假设下建立中心 L 值的 Ichino-Ikeda 型精细公式；同时证明其分裂类比 (GL_{2n},GL_1) 的相应猜想与精细形式。
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_n\times U_{n+1}$ in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called $*$-generic, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. The other, built upon Zeta integrals, is expressed in terms of fu…
Non-tempered Gan-Gross-Prasad conjecture for Archimedean general linear groups
本文针对阿基米德局部域上的一般线性群，证明了非tempered Gan-Gross-Prasad猜想中“周期蕴含相关”的方向，结合已有结果解决了该猜想。
In this paper, we investigate the local Gan-Gross-Prasad conjecture for some pair of representations of $U(3)\times U(2)$ involving a non-generic representation. For a pair of generic $L$-parameters of $(U(n),U(n-1))$, it is known that there is a unique pair of representations in their associateed Vogan $L$-packets which produces the unique Bessel model of these $L$-parameters. We showed that this is not ture for some pair of $L$-parameters involving a non-generic one. On the other hand, we give the precise local theta correspondence for $(U(1),U(3))$ not at the level of $L$-parameters but of …
Let F be a number field, A_F its ring of adeles, and let π_n and π_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to π_n and π_{n+1} is equivalent to the non-vanishing of L(1/2, π_n x π_{n+1}). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period int…
Duality for generalized Gan-Gross-Prasad relevant pairs for $p$-adic $\mathrm{GL}_n$
本文提出广义 GGP 相关对概念以描述 p 进一般线性群的商分支律，并基于导数与积分的交换关系，从表示论和组合角度证明相关对的对偶性。
We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The…
On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture
本文通过组合多个theta对应，在（SO(5), SO(2)）情形下证明了Bessel周期的Ichino-Ikeda型公式，并由此将Böcherer猜想推广到非平凡环面特征情形。
Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
本文利用 Bessel 模型为 $\mathrm{GSpin}$ 群构造了张量积部分 $L$-函数乘积的 Rankin-Selberg 积分，并以此证明了拟分裂 $\mathrm{GSpin}$ 群一般表示的全局 Gan-Gross-Prasad 猜想的一个方向。
Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space Hom_{H(F)}(π_{| H(F)},σ). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that…
We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.
Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture
本文在特殊贝塞尔周期情形下证明了刘一峰提出的精细全局Gross-Prasad猜想，建立了中心L值的Ichino-Ikeda型显式公式，并由此证明了关于度为二的Hecke特征形式全纯Siegel尖点形式的Boecherer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。
Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs
本文针对非阿基米德局部域上的GL_{n+1}与GL_n，给出并证明了二者不可约光滑表示间同态非零的充要条件，解决了商分支律问题，还将结果推广到A型仿射Hecke代数并细化了分支律。
In \cite{JZ1}, D. Jiang and L. Zhang proposed a conjecture which related the wavefront sets and the descent method in the local fields case. Recently, in \cite{JLZ}, they and D. Liu define the arithmetic wavefront set of certain irreducible admissible representation $π$ of a classical group $G(k)$ defined over local field $k$, which is a subset of $k$-rational nilpotent orbits of the Lie algebra of $G(k)$, by the arithmetic structures of the enhanced L-parameter of $π$. These arithmetic structures are based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad…
In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.
Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.
Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality
本文为经典群选取与抛物诱导和theta对应相容的典范Lusztig对应，推广Pan的结果，并证明theta对应与有限Gan-Gross-Prasad问题的对偶，作为相对Langlands对偶的有限域实例。
Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.
A note on toric periods in unramified families
本文研究了在无分支族中托里周期的规范问题，解决了D.Prasad提出的问题并指出了更广泛应用的可能性。
In this paper, we give an explicit determination of the non-vanishing of the theta liftings for unitary dual pairs (U(p,q), U(r,s)). Assuming the local Gan-Gross-Prasad conjecture, we determine when theta lifts of tempered representations are nonzero in terms of the local Langlands correspondence.
In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $Π$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representati…
Fourier-Jacobi models for real symplectic-metaplectic groups: the basic case
本文推广了Gan-Ichino和Atobe的方法，证明了傅里叶-雅可比模型下辛-元摄引群局部Gan-Gross-Prasad猜想的基本温顺情形。
The relative trace formula of Jacquet-Rallis (for unitary groups or general linear groups) is an identity between periods of automorphic representations and geometric distributions. In this paper, we prove the transfer between all geometric terms of unitary groups and those of linear groups. We also show that all geometric terms are in the weak closure of local regular semi-simple orbital integrals. We mention an application to the Gan-Gross-Prasad conjecture for unitary groups.
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