Avraham Aizenbud, Dmitry Gourevitch
In this paper we prove that the symmetric pair $(GL_{n+k}(F),GL_n(F) \times GL_k(F))$ is a Gelfand pair for any local field F of characteristic 0. For non-archimedean F it has been proven in [JR]. We use techniques developed in [AG2] to generalize their proof to general local fields.
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On the geometric side of the Jacquet-Rallis relative trace formula
本文研究了 Jacquet-Rallis 相对迹公式的几何侧，全局计算了一般线性群上的几何项并证明其由正则轨道积分描述，局部证明了正则轨道积分与酉群半单轨道积分可比较。
We give a new proof of the so-called Lie algebra version of Jacquet-Rallis's fundamental lemma for local non-Archimedean fields of characteristic zero. This proof is local and based on a previous result of W. Zhang on the compatibility of smooth transfer with a (partial) Fourier transform.
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.
We establish the coarse relative trace formulae of Jacquet-Rallis for linear and unitary groups. Both formulae are of the form: a sum of spectral distributions equals a sum of geometric distributions. In order to obtain the spectral decompositions we introduce new truncation operators and we investigate their properties. On the geometric side, by means of the Cayley transform, the decompositions are derived from a procedure of descent to the tangent spaces for which the formulae are known thanks to our previous work.
Gaussian test functions and Jacquet-Rallis transfer
本文利用Kudla-Millson形式体系与对称空间几何性质，为Jacquet-Rallis相对迹公式比较的一般线性侧构造高斯测试函数，并给出其向紧酉群转移及Howe算子显式公式。
In this paper, we prove the uniqueness of Ginzburg-Rallis models in the archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent $χ$-incompatibility.
Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case
本文通过乘法希钦纤维化方法，证明了局部函数域上球面海加以尔代数的雅各布森-拉里斯基本引理。
In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the se…
We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power $s \in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s, \cdot)$ indexed by the invariants of the adjoint action of $\mathrm{GL}_n(\mathrm{F})$ on $\mathfrak{gl}_{n+1}(\mathrm{F})$ as well as a "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mat…
The Uniqueness of the Ginzburg-Rallis Model: the Non-Archimedean Case
本文证明了特征零p-进局部域上Ginzburg-Rallis模型的唯一性，通过推广阿基米德情形策略并利用𝔷-有限分布波前集的精细结构，完成了该模型局部唯一性问题的最后一步。
Jacquet--Rallis transfer for GL(2)
该论文研究 GL(2) 的 Jacquet--Rallis 转移，证明了在 n=2 情形下多项式型 Schwartz 函数在轨道积分间的双向转移存在性，并提出了相关 (g,K)-模的结构猜想。
On the trilinear and Ginzburg-Rallis models
本文给出非阿基米德局部域上GL_6(k)的抛物诱导表示中Ginzburg-Rallis模型与诱导数据的三线性模型同构的充分条件，并给出这些模型非零的判别准则。
In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.
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. In this paper we explicitly calculate archimedean zeta integrals of this type for U(n,1), assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.
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…
We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands corresponden…
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.
Given a reductive group $G$ and a reductive subgroup $H$, both defined over a number field $F$, we introduce the notion of the $H$-distinguished automorphic spectrum of $G$ and analyze it for the pairs $(GL_{2n},Sp_n)$ and $(Sp_{2n},Sp_n\times Sp_n)$. In the first case we give a complete description using results of Jacquet--Rallis, Offen and Yamana. In the second case we give an upper bound, generalizing vanishing results of Ash--Ginzburg--Rallis and a lower bound, extending results of Ginzburg--Rallis--Soudry.
Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$
本文研究了$GL_{2n}(F)$主级数表示的扭曲Jacquet模块结构，给出了非零条件，并探讨了非阿基米德情况下的Shalika模型存在性。
We prove the functional equation of the non archimedean exterior-square L-function of irreducible representations of GL(n), when n is odd.
On the Jacquet functor of Symplectic groups
本文证明，对于非阿基米德局部域上辛群Sp(2n,F)的不可约光滑表示的一个等价类，关于极大Levi子群GL(l,F)×Sp(2n-2l,F)的Jacquet函子是无重数的。
This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a non-Archimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein's presentation of the Iwahori-Hecke algebra.
The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to our recent work in the reall case joint with C. Cheng and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation $π$ of $\mathrm{GL}_{2n}(\mathbb{C})$ with a non-zero Shalika model; (2) construct a new twisted linear period $Λ_{s,χ}$; (3) give a necessary and sufficient condition on the character $χ$ such that there exists a uniform cohomological test vector $v\in V_π$…
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of GL(n) over a non-archimedean local field by means of the local Jacquet-Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.
Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation π of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, π corresponds to an irreducible representation ρ of the multiplicative group of the central division algebra over F with invariant 1/…
Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(π),ρ)<= 1, for any irreducible representations πof G and ρof M. To do that we adapt the classical method of Ge…
Local newforms and spherical characters for unitary groups
本文证明了类似Jacquet-Rallis基本引理的平滑转移，计算了Ichino-Ikeda猜想中的局部球面特征标，并推出酉群局部新形式的存在性。
Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of $B^\times$ of dimension $>1$ and irreducible cuspidal representations of $GL_2(F)$. We present a new construction of this bijection in which the preservation of epsilon factors is automatic.
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.
Let $F$ be a non archimedean local field, and $n_1$ and $n_2$ two positive even integers. We prove that if $π_1$ and $π_2$ are two smooth representations of $GL(n_1,F)$ and $GL(n_2,F)$ respectively, both admitting a Shalika period, then the normalised parabolically induced representation $π_1\times π_2$ also admits a Shalika period. Combining this with the results of \cite{M-localBF}, we obtain as a corollary the classification of generic representations of $GL(n,F)$ admitting a Shalika period when $F$ has characteristic zero. This result is relevant to the study of the Jacquet-Shalika exterio…
Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series
本文证明 Kudla–Rallis 关于退化主级数表示在 G(W)×G(W) 上限制后 Hom 空间维数为 1 的猜想，并推广到正交、酉情形，从而重新证明局部 Theta 对应守恒关系。
Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).
Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for $G$ can be described using admissible embeddings of L-tori.
We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo-Jacquet trace formulae and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe's finiteness for weighted orbital integrals and the representability of the Fourier transform of weighted orbital integrals. These local results are prepared for the comparison of regular semi-simple terms, which are weig…
Local Jacquet-Shalika integrals and modifying factors
本文在特征零的局部域上发展Jacquet-Shalika积分的局部理论，建立匹配Artin局部ε-因子的完整函数方程，核心采用与Shalika子群相关开轨道积分的新比较策略。
Let $F$ be a non-archimedean local field. For any irreducible representation $π$ of an inner form $G'=\mathrm{GL}_{m}(D)$ of $G=\mathrm{GL}_{N}(F)$, there exists an irredubile representation of a maximal compact open subgroup in $G'$ which is also a type for $π$. Then we can consider the problem whether these types are unique or not in some sense. If such types for $π$ are unique, we say $π$ has the strong unicity property of types. On the other hand, there exists a correspondence connecting irreducible representations of $G'$ and $G$, called the Jacquet--Langland correspondence. In this paper…
The Local Converse Problem is to determine how the family of the local gamma factors $γ(s,π\timesτ,ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, w…
Finite period vectors and Gauss sums
本文通过显式计算与 Deligne–Kazhdan 理论，建立有限域及局部域上多种 gamma 因子与 Gauss 和、Artin 因子的关系，并刻画相关周期向量。
In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.
Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$
本文研究非阿基米德局部域上辛群$Sp_4(F)$的主系列表示，确定了由秩1二次型特征标给出的扭曲Jacquet模作为稳定子群模的结构，并推广到有限域情形。
Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earl…
Suppose $F$ is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on $n \times n$ matrices $M_n(F)$ to define the local standard $L$-functions on $\mathrm{GL}_n$. The purpose of this partly expository note is to give evidence that there is an analogous and useful "approximate" Godement-Jacquet theory for the standard $L$-functions on the special orthogonal groups $\mathrm{SO}(V)$: One replaces $\mathrm{GL}_n(F)$ with $\mathrm{GSpin}(V)(F)$ and $M_n(F)$ with $\mathrm{Clif}(V)(F)$, the Clifford algebra of $V$. More precisely, we exp…
In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of redu…
Finite-Sum Realization of Archimedean Asai and Exterior-Square $L$-Factors
该研究在数论领域，将 Archimedean Asai L-因子与外平方 L-因子分别表示为特定局部 ζ 积分的有限和，完善了 L-因子的有限和实现相关理论。
Archimedean Newform Theory for $\mathrm{GL}_n$
本文为 $\mathrm{GL}_n$ 的 Archimedean Casselman-Wallach 表示定义导子指数并构造唯一性新形式，证明其在 Rankin-Selberg 积分中可作测试向量，从而补全全局新形式理论。
We calculate the asymptotic behavior of the dimension of the fixed vectors of $π$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $π$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $π$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.
Godement--Jacquet L-function and homological theta lifting
本文结合同调与解析方法，研究非阿基米德局部域上 II 型对偶对的 theta 提升，完全确定了临界点处 L-函数全纯的不可约表示的大 theta 提升，计算了所有特征标的大 theta 提升，并给出了 Weil 表示射影性的充要条件。
Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations $π$ of GL_n(F). In this paper, we show that Jacquet-Shalika integral attains a certain L-function, so called the formal exterior square L-function, when the Whittaker function is associated to a newform for $π$. By consideration on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.
Archimedean Bernstein-Zelevinsky Theory and Homological Branching Laws
本文发展拟分裂实经典群的 Bernstein-Zelevinsky 理论，建立一般线性群的 Euler-Poincaré 特征标公式，并证明高阶扩张群消失、最高导数 Leibniz 律、酉性判据及分裂偶正交群扭曲同调的 Hausdorff 性与正合性。
On modular representations of inner forms of $\mathrm{GL}_n$ over a local non-archimedean field
该研究将局部非阿基米德域上$\n\thrm{GL}_n$内形式的$\nsquare$-不可约表示结果推广到模系数域，给出$L$-因子计算、分类穷举性新证明及抛物诱导表示成分刻画。
Rationality of the Local Jacquet-Langlands Correspondence for GL(n)
本文研究局部 Jacquet-Langlands 对应中表示与其 L-参数的定义域关系，通过有理域上的除环和 Hasse 不变量刻画该对应，并确定非 p 位及附加条件下 p 位的不变量。
We prove that the Jacquet restriction functor for a parabolic subgroup of a reductive group over a non-Archimedean local field is right adjoint to the parabolic induction functor for the opposite parabolic subgroup, in the generality of smooth group representations on R-modules for any unital ring R in which the residue field characteristic is invertible. Correction: it turned out that the paper contains a mistake (in Lemma 3.4), which the authors have been unable to fix. This renders the proof of the main theorem incomplete.
Rallis and Soudry have proven the stability under twists by highly ramified characters of the local gamma factor arising from the doubling method, in the case of a symplectic group or orthogonal group G over a local non-archimedean field F of characteristic zero, and a representation of G, which is not necessarily generic. This paper extends their arguments to show the stability in the case when G is a unitary group over a quadratic extension E of F, thereby completing the proof of the stability for classical groups. This stability property is important in Cogdell, Piatetski-Shapiro, and Shahi…
Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying ou…
Let F be a non-archimedean local field of characteristic zero. Jacquet, Piatetski-Shapiro and Shalika introduced the notion of newforms for irreducible generic representations of GL_n(F). In this paper, we give an explicit formula for Whittaker functions associated to newforms on the diagonal matrices in GL_n(F).
A spectral expansion for the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$
本文证明数域二次扩张下对称空间GL_n(E)/GL_n(F)上theta级数的谱展开，通过推广intertwining periods至离散自守表示并建立离散Eisenstein级数在虚轴邻域的界，为Jacquet-Rallis迹公式精细谱展开提供关键步骤。
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。
Let F be a non-archimedean local field of characteristic zero with residual characteristic p. In this paper, we present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of GL_n(F), when p does not divide n. As an application, we prove Jacquet's conjecture on the local converse problem for GL_n(F) in the case of simple supercuspidal representations, for arbitrary p.
On the product functor on inner forms of the general linear group over a non-Archimedean local field
本文证明了在非阿基米德一般线性群内形式的光滑表示范畴的全子范畴上，抛物诱导函子是完全忠实的，并应用该结果证明由 Jacquet 函子产生的特定大导数具有不可约基座且重数为 1。
In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. W…
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