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The existence of positive almost periodic type solutions for some neutral nonlinear integral equation

2004-12-09
Bin Xu , Rong Yuan

On the positive almost periodic type solutions for some nonlinear delay integral equations

2004-12-06
Bin Xu , Rong Yuan

On the cuspidal support of discrete series for $p$-adic quasisplit $Sp(N)$ and $SO(N)$

2015-01-01
Bin Xu
Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. MƓglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is 
 Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. MƓglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of MƓglin and Tadic in the case of quasisplit symplectic groups and orthogonal groups by using Arthur's theory.
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Classification, reduction, group invariant solutions and conservation laws of the Gardner-KP equation

2009-07-02
Bin Xu , Xi-Qiang Liu

Arthur packets for 𝑝-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

2022-03-01
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
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The Jacquet–Langlands Correspondence via Twisted Descent

2015-10-29
Dihua Jiang , Baiying Liu , Bin Xu +1 more
The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained 
 The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972 [30]. In this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry [10] to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence for |${\mathrm {PGL}}(2)$| via a new explicit construction.
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Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

2017-05-04
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

2018-05-04
Suting Wei , Bin Xu , Jun Yang
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Residues of Intertwining Operators for <i>U</i>(3,3) and Base Change

2014-04-25
Li Cai , Bin Xu
Abstract Let F be a p-adic local field with characteristic 0 and E/F a quadratic extension. Let σ=πG⊗πH be a supercuspidal representation of GL2(E)×UE/F(1,1) with πG conjugate self-dual. In this 
 Abstract Let F be a p-adic local field with characteristic 0 and E/F a quadratic extension. Let σ=πG⊗πH be a supercuspidal representation of GL2(E)×UE/F(1,1) with πG conjugate self-dual. In this paper, we use a local approach developed by Shahidi to study the residue of the standard intertwining operator on the parabolic induced representation from σ to the quasi-split unitary group UE/F(3,3) at s=0, and show that if πG is a standard base change from UE/F(1,1), then the residue at s=0 is nonzero if and only if πG is the standard base change of πH.

Arthur packets and Adams-Barbasch-Vogan packets for p-adic groups, 1: Background and Conjectures

2019-06-04
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

On top Fourier coefficients of certain automorphic representations of GLn

2018-01-01
Baiying Liu , Bin Xu
In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations 
 In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ of form $$ \Delta(\tau_1, b_1) \boxplus \Delta(\tau_2, b_2) \boxplus \cdots \boxplus \Delta(\tau_r, b_r)\,, $$ where $\Delta(\tau_i,b_i)$'s are Speh representations in the discrete spectrum of $\mathrm{GL}_{a_ib_i}(\mathbb{A})$ with $\tau_i$'s being unitary cuspidal representations of $\mathrm{GL}_{a_i}(\mathbb{A})$, and $n = \sum_{i=1}^r a_ib_i$. Endoscopic lifting images of the discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.
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A reciprocal branching problem for automorphic representations and global Vogan packets

2019-08-14
Dihua Jiang , Baiying Liu , Bin Xu
Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G 
 Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G , determine the occurrence of an irreducible representation σ of H in the restriction of π to H . The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H , find an irreducible representation π of G such that σ occurs in the restriction of π to H . For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.
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The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

2005-09-27
Bin Xu , Rong Yuan

On top Fourier coefficients of certain automorphic representations of $${\mathrm {GL}}_n$$

2020-01-09
Baiying Liu , Bin Xu

Isolation of the cuspidal spectrum: The function field case

2021-07-18
Li Cai , Bin Xu
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The Jacquet Langlands correspondence via twisted descent

2015-01-01
Dihua Jiang , Baiying Liu , Bin Xu +1 more
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by 
 The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
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On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains

2016-01-01
Suting Wei , Bin Xu , Jun Yang
We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is 
 We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni(p.327, [4]).
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New Solutions of the Generalized Burgers-KP Equation

2009-01-01
Bin Xu
Three types of travelling solutions of the Burgers-KP equations are obtained via G'/G-expansion method and it was taken to the generalized Burgers-KP with variable-coefficient,furthermore the corresponding results given by Wazwaz 
 Three types of travelling solutions of the Burgers-KP equations are obtained via G'/G-expansion method and it was taken to the generalized Burgers-KP with variable-coefficient,furthermore the corresponding results given by Wazwaz are generalized,and some new travelling wave solutions of the generalized Burgers-KP with variable-coefficient are found out.

A reciprocal branching problem for automorphic representations and global Vogan packets

2018-01-01
Dihua Jiang , Baiying Liu , Bin Xu
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $\pi$ of $G$, determine the occurrence 
 Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $\pi$ of $G$, determine the occurrence of an irreducible representation $\sigma$ of $H$ in the restriction of $\pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $\sigma$ of $H$, find an irreducible representation $\pi$ of $G$ such that $\sigma$ occurs in the restriction of $\pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
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On Local Descent to Metaplectic Groups and Local Theta Correspondence

2020-04-13
Bin Xu

The characteristic cycles and semi-canonical bases on type $A$ quiver variety

2020-01-01
Taiwang Deng , Bin Xu
In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical 
 In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
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Isolation of the Cuspidal Spectrum: the Function Field Case

2021-01-01
Li Cai , Bin Xu
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test 
 Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case.
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The characteristic cycles and semi-canonical bases on type A quiver variety

2022-02-02
Taiwang Deng , Bin Xu
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Interfaces with boundary intersection for an inhomogeneous Allen-Cahn equation in three-dimensional case

2023-01-01
Qiang Ren , Bin Xu
The following boundary value problem is considered: \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} \varepsilon^2\Delta u+V(y)u(1-u^2) = 0, &amp;\mbox{in}~~ \Omega , \\ \frac{\partial u}{\partial n} = 0, &amp;\mbox{on }~~ \partial \Omega, \end{array}\right. \end{equation*} $\end{document} 
 The following boundary value problem is considered: \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} \varepsilon^2\Delta u+V(y)u(1-u^2) = 0, &amp;\mbox{in}~~ \Omega , \\ \frac{\partial u}{\partial n} = 0, &amp;\mbox{on }~~ \partial \Omega, \end{array}\right. \end{equation*} $\end{document} where $ \Omega\subset \mathbb R^3 $ is a smooth bounded domain. Assume that $ \Gamma\subset \bar{\Omega} $ is a smooth surface which intersects $ \partial\Omega $ at a right angle and separates $ \Omega $ into two parts $ \Omega_1 $ and $ \Omega_2 $. In addition, we also assume that $ \Gamma $ is a non-degenerate critical point of the functional $ \mathcal{K}(\Gamma) = \int_\Gamma V^{\frac12}d\mu $. From the infinite dimensional reduction method, we find a particular kind of solutions which converge to $ 1 $ in $ \Omega_1 $ and to $ -1 $ in $ \Omega_2 $ as $ \varepsilon\to 0 $.
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Automorphic descent for symplectic groups: The branching problems and L -functions

2023-06-01
Baiying Liu , Bin Xu
We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of 
 We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for ${\rm Mp}_2(\Bbb{A})$, we give a new approach to prove the result that for an automorphic cuspidal representation of ${\rm GL}_2(\Bbb{A})$ of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central $L$-values.

Global L -Packets of Quasisplit GSp(2 n ) and GO(2 n )

2025-03-27
Bin Xu
abstract: This is a sequel to Xu (2018) on the $L$-packets of quasisplit general symplectic and even orthogonal groups. We show the existence of global $L$-packets and establish the functoriality 
 abstract: This is a sequel to Xu (2018) on the $L$-packets of quasisplit general symplectic and even orthogonal groups. We show the existence of global $L$-packets and establish the functoriality of endoscopic transfer for them in many cases.

The small value set polynomials over finite fields and monodromy groups

2025-05-01
Xiantao Deng , Bin Xu , Quanmin Zhu

The small value set polynomials over finite fields and monodromy groups

2025-05-01
Xiantao Deng , Bin Xu , Quanmin Zhu

Global L -Packets of Quasisplit GSp(2 n ) and GO(2 n )

2025-03-27
Bin Xu
abstract: This is a sequel to Xu (2018) on the $L$-packets of quasisplit general symplectic and even orthogonal groups. We show the existence of global $L$-packets and establish the functoriality 
 abstract: This is a sequel to Xu (2018) on the $L$-packets of quasisplit general symplectic and even orthogonal groups. We show the existence of global $L$-packets and establish the functoriality of endoscopic transfer for them in many cases.

Automorphic descent for symplectic groups: The branching problems and L -functions

2023-06-01
Baiying Liu , Bin Xu
We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of 
 We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for ${\rm Mp}_2(\Bbb{A})$, we give a new approach to prove the result that for an automorphic cuspidal representation of ${\rm GL}_2(\Bbb{A})$ of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central $L$-values.

Interfaces with boundary intersection for an inhomogeneous Allen-Cahn equation in three-dimensional case

2023-01-01
Qiang Ren , Bin Xu
The following boundary value problem is considered: \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} \varepsilon^2\Delta u+V(y)u(1-u^2) = 0, &amp;\mbox{in}~~ \Omega , \\ \frac{\partial u}{\partial n} = 0, &amp;\mbox{on }~~ \partial \Omega, \end{array}\right. \end{equation*} $\end{document} 
 The following boundary value problem is considered: \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} \varepsilon^2\Delta u+V(y)u(1-u^2) = 0, &amp;\mbox{in}~~ \Omega , \\ \frac{\partial u}{\partial n} = 0, &amp;\mbox{on }~~ \partial \Omega, \end{array}\right. \end{equation*} $\end{document} where $ \Omega\subset \mathbb R^3 $ is a smooth bounded domain. Assume that $ \Gamma\subset \bar{\Omega} $ is a smooth surface which intersects $ \partial\Omega $ at a right angle and separates $ \Omega $ into two parts $ \Omega_1 $ and $ \Omega_2 $. In addition, we also assume that $ \Gamma $ is a non-degenerate critical point of the functional $ \mathcal{K}(\Gamma) = \int_\Gamma V^{\frac12}d\mu $. From the infinite dimensional reduction method, we find a particular kind of solutions which converge to $ 1 $ in $ \Omega_1 $ and to $ -1 $ in $ \Omega_2 $ as $ \varepsilon\to 0 $.
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Arthur packets for 𝑝-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

2022-03-01
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
View PDF Chat

The characteristic cycles and semi-canonical bases on type A quiver variety

2022-02-02
Taiwang Deng , Bin Xu
View PDF Chat

Isolation of the cuspidal spectrum: The function field case

2021-07-18
Li Cai , Bin Xu
View PDF Chat

Isolation of the Cuspidal Spectrum: the Function Field Case

2021-01-01
Li Cai , Bin Xu
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test 
 Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case.
View PDF Chat

On Local Descent to Metaplectic Groups and Local Theta Correspondence

2020-04-13
Bin Xu

On top Fourier coefficients of certain automorphic representations of $${\mathrm {GL}}_n$$

2020-01-09
Baiying Liu , Bin Xu

The characteristic cycles and semi-canonical bases on type $A$ quiver variety

2020-01-01
Taiwang Deng , Bin Xu
In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical 
 In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
View PDF Chat

A reciprocal branching problem for automorphic representations and global Vogan packets

2019-08-14
Dihua Jiang , Baiying Liu , Bin Xu
Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G 
 Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G , determine the occurrence of an irreducible representation σ of H in the restriction of π to H . The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H , find an irreducible representation π of G such that σ occurs in the restriction of π to H . For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.
View PDF Chat

Arthur packets and Adams-Barbasch-Vogan packets for p-adic groups, 1: Background and Conjectures

2019-06-04
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

2018-05-04
Suting Wei , Bin Xu , Jun Yang
View PDF Chat

On top Fourier coefficients of certain automorphic representations of GLn

2018-01-01
Baiying Liu , Bin Xu
In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations 
 In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ of form $$ \Delta(\tau_1, b_1) \boxplus \Delta(\tau_2, b_2) \boxplus \cdots \boxplus \Delta(\tau_r, b_r)\,, $$ where $\Delta(\tau_i,b_i)$'s are Speh representations in the discrete spectrum of $\mathrm{GL}_{a_ib_i}(\mathbb{A})$ with $\tau_i$'s being unitary cuspidal representations of $\mathrm{GL}_{a_i}(\mathbb{A})$, and $n = \sum_{i=1}^r a_ib_i$. Endoscopic lifting images of the discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.
View PDF Chat

A reciprocal branching problem for automorphic representations and global Vogan packets

2018-01-01
Dihua Jiang , Baiying Liu , Bin Xu
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $\pi$ of $G$, determine the occurrence 
 Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $\pi$ of $G$, determine the occurrence of an irreducible representation $\sigma$ of $H$ in the restriction of $\pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $\sigma$ of $H$, find an irreducible representation $\pi$ of $G$ such that $\sigma$ occurs in the restriction of $\pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
View PDF Chat

Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

2017-05-04
Clifton Cunningham , Andrew Fiori , Ahmed Moussaoui +2 more
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, 
 In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains

2016-01-01
Suting Wei , Bin Xu , Jun Yang
We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is 
 We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni(p.327, [4]).
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The Jacquet–Langlands Correspondence via Twisted Descent

2015-10-29
Dihua Jiang , Baiying Liu , Bin Xu +1 more
The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained 
 The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972 [30]. In this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry [10] to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence for |${\mathrm {PGL}}(2)$| via a new explicit construction.
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The Jacquet Langlands correspondence via twisted descent

2015-01-01
Dihua Jiang , Baiying Liu , Bin Xu +1 more
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by 
 The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
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On the cuspidal support of discrete series for $p$-adic quasisplit $Sp(N)$ and $SO(N)$

2015-01-01
Bin Xu
Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. MƓglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is 
 Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. MƓglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of MƓglin and Tadic in the case of quasisplit symplectic groups and orthogonal groups by using Arthur's theory.
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Residues of Intertwining Operators for <i>U</i>(3,3) and Base Change

2014-04-25
Li Cai , Bin Xu
Abstract Let F be a p-adic local field with characteristic 0 and E/F a quadratic extension. Let σ=πG⊗πH be a supercuspidal representation of GL2(E)×UE/F(1,1) with πG conjugate self-dual. In this 
 Abstract Let F be a p-adic local field with characteristic 0 and E/F a quadratic extension. Let σ=πG⊗πH be a supercuspidal representation of GL2(E)×UE/F(1,1) with πG conjugate self-dual. In this paper, we use a local approach developed by Shahidi to study the residue of the standard intertwining operator on the parabolic induced representation from σ to the quasi-split unitary group UE/F(3,3) at s=0, and show that if πG is a standard base change from UE/F(1,1), then the residue at s=0 is nonzero if and only if πG is the standard base change of πH.

Classification, reduction, group invariant solutions and conservation laws of the Gardner-KP equation

2009-07-02
Bin Xu , Xi-Qiang Liu

New Solutions of the Generalized Burgers-KP Equation

2009-01-01
Bin Xu
Three types of travelling solutions of the Burgers-KP equations are obtained via G'/G-expansion method and it was taken to the generalized Burgers-KP with variable-coefficient,furthermore the corresponding results given by Wazwaz 
 Three types of travelling solutions of the Burgers-KP equations are obtained via G'/G-expansion method and it was taken to the generalized Burgers-KP with variable-coefficient,furthermore the corresponding results given by Wazwaz are generalized,and some new travelling wave solutions of the generalized Burgers-KP with variable-coefficient are found out.

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

2005-09-27
Bin Xu , Rong Yuan

The existence of positive almost periodic type solutions for some neutral nonlinear integral equation

2004-12-09
Bin Xu , Rong Yuan

On the positive almost periodic type solutions for some nonlinear delay integral equations

2004-12-06
Bin Xu , Rong Yuan
Coauthor Papers Together
Baiying Liu 7
Dihua Jiang 4
Clifton Cunningham 3
Andrew Fiori 3
James Mracek 3
Rong Yuan 3
Ahmed Moussaoui 3
Suting Wei 2
Taiwang Deng 2
Li Cai 2
Jun Yang 2
Lei Zhang 1
Lei Zhang 1
Xi-Qiang Liu 1
Xiantao Deng 1
Qiang Ren 1
Li Cai 1
Quanmin Zhu 1

The Endoscopic Classification of Representations

2013-10-31
James Arthur
Table of Contents:* Parameters * Local transfer * Global stabilization * The standard model * A study of critical cases * The local classification * Local nontempered representations * The 
 Table of Contents:* Parameters * Local transfer * Global stabilization * The standard model * A study of critical cases * The local classification * Local nontempered representations * The global classification * Inner forms * Bibliography * Index

A Proof of Langlands' Conjecture on Plancherel Measures; Complementary Series of -adic groups

1990-09-01
Freydoon Shahidi
analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at 
 analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

Multiplicity one theorems

2010-01-01
Avraham Aizenbud , Dmitry Gourevitch , Stephen Rallis +1 more
In the local, characteristic 0, non-Archimedean case, we consider distributions on GL(n + 1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant 
 In the local, characteristic 0, non-Archimedean case, we consider distributions on GL(n + 1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies multiplicity at most one for restrictions from GL(n + 1) to GL(n). Similar theorems are obtained for orthogonal or unitary groups.
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Uniqueness of Bessel Models: The Archimedean Case

2010-08-08
Dihua Jiang , Binyong Sun , Chen-Bo Zhu
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Fourier coefficients for automorphic forms on quasisplit classical groups

2016-01-01
Dihua Jiang , Baiying Liu
In (J14), a conjecture was proposed on a relation be- tween the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the correspond- ing global 
 In (J14), a conjecture was proposed on a relation be- tween the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the correspond- ing global Arthur packets. In this paper, we discuss the recent progress on this conjecture and certain problems which lead to better understanding of Fourier coefficients of automorphic forms. At the end, we extend a useful technical lemma to a few versions, which are more convenient for future applications.
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Automorphic Forms on GL (2)

1970-01-01
H. Jacquet , R. P. Langlands
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Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

2009-01-01
Wee Teck Gan , Benedict H. Gross , Dipendra Prasad
We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. 
 We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

2014-01-01
Tasho Kaletha , Alberto MĂ­nguez , Sug Woo Shin +1 more
We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is 
 We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.

Induced representations of reductive ${\germ p}$-adic groups. I

1977-01-01
I. N. Bernstein , Andrei Zelevinsky
© Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1977, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions 
 © Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1977, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.
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The local converse theorem for SO(2n + 1) and applications

2003-05-01
Dihua Jiang , David Soudry
In this paper we characterize irreducible generic representations of SO 2n+1 (k) (where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem).As applications, 
 In this paper we characterize irreducible generic representations of SO 2n+1 (k) (where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem).As applications, we prove that two irreducible generic cuspidal automorphic representations of SO 2n+1 (A) (where A is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of SO 2n+1 (k).
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On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups

2014-03-25
Dihua Jiang , Baiying Liu
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Bifurcation of almost periodic solutions for a nonlinear integral equation with delay

1996-10-01
S. Chen , Ricardo TorrejĂłn

Nonzero Solutions of Nonlinear Integral Equations Modeling Infectious Disease

1982-01-01
Lynn R. Williams , Richard W. Leggett
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, $x(t) = \int_{t - \tau }^t {f(s,x(s))ds} $, are given in terms of simple product and 
 Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, $x(t) = \int_{t - \tau }^t {f(s,x(s))ds} $, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if $x(t)$ is the proportion of infectives at time t and $f(t,x(t))$ is the proportion of new infectives per unit time.

Almost Periodic Differential Equations

1974-01-01
A. M. Fink
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Positive almost periodic solutions of some delay integral equations

1990-01-01
A. M. Fink , Juan A. Gatica

Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés

2001-01-01
Jean-Loup Waldspurger

Endoscopic Classification of representations of Quasi-Split Unitary Groups

2014-10-21
Chung Pang Mok
Introduction Statement of the main theorems Local character identities and the intertwining relation Trace formulas and their stabilization The Standard model Study of critical cases Local classification Nontempered representations Global 
 Introduction Statement of the main theorems Local character identities and the intertwining relation Trace formulas and their stabilization The Standard model Study of critical cases Local classification Nontempered representations Global classification Addendum Bibliography
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Equivariant Sheaves and Functors

1994-01-01
Joseph Bernstein , Valery A. Lunts
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Raising nilpotent orbits in wave-front sets

2016-10-26
Dihua Jiang , Baiying Liu , Gordan Savin
We study wave-front sets of representations of reductive groups over global or non-archimedean local fields. We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.
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Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains

2013-07-05
Ying Guo , Jun Yang
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Multiplicity one theorems: the Archimedean case

2011-12-07
Binyong Sun , Chen-Bo Zhu
Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, 
 Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\GL_{n}(\R)\ltimes \oH_{2n+1}(\R)$, $\GL_{n}(\C)\ltimes \oH_{2n+1}(\C)$, $\oU(p,q)\ltimes \oH_{2p+2q+1}(\R)$, $\Sp_{2n}(\R)\ltimes \oH_{2n+1}(\R)$, $\Sp_{2n}(\C)\ltimes \oH_{2n+1}(\C)$, with their respective subgroups $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\Sp_{2n}(\R)$, $\Sp_{2n}(\C)$.
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Eisenstein Series and Automorphic 𝐿-Functions

2010-09-22
Freydoon Shahidi

Rankin-Selberg Convolutions

1983-04-01
H. Jacquet , I. I. Piatetskii-Shapiro , J. A. Shalika

Poles of certain residual Eisenstein series of classical groups

2013-07-05
Dihua Jiang , Baiying Liu , Lei Zhang
In memory In memory
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Pseudo Almost Periodic Solutions of Some Differential Equations

1994-01-01
C.Y. Zhang

Generalized and degenerate Whittaker models

2017-02-01
Raul Gomez , Dmitry Gourevitch , Siddhartha Sahi
We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized 
 We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$ . Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$ , see Moeglin and Waldspurger [ Modeles de Whittaker degeneres pour des groupes p-adiques , Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I. , Ann. Sci. Éc. Norm. SupĂ©r. (4) 10 (1977), 441–472; Aizenbud et al. , Derivatives for representations of $\text{GL}(n,\mathbb{R})$ and $\text{GL}(n,\mathbb{C})$ , Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.
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Local Δ-Factors and Characters of GL(2)

1983-12-01
Jerrold B. Tunnell
Introduction. The representation theory of GL(2, K), for K a nonarchimedean local field, is somewhat more complicated than the corresponding theory for GL(2, R). In particular, the formulas for the 
 Introduction. The representation theory of GL(2, K), for K a nonarchimedean local field, is somewhat more complicated than the corresponding theory for GL(2, R). In particular, the formulas for the characters of discrete series representation of GL(2, R) have a concise closed form, while the published character formulas for supercuspidal representations of GL(2, K) occupy several pages [10], [14]. The purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of GL(2, K). The representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in GL(2, K), and the subspace stabilized by an open compact subgroup of GL(2, K) is finite dimensional. Irreducible admissible representations ir of GL(2, K) satisfy Schur's lemma, so that the center K* of GL(2, K) acts by multiplication by a quasicharacter ,. There exists a locally constant character function ch , defined on a dense open subgroup of GL(2, K), which determines the isomorphism class of ir [7, Section 7]. Jacquet and Langlands [7, Section 2] introduced factors E(10 X) attached to the twist of ir by quasicharacters X of K* which describe the action of -r in a specific model. For each separable quadratic extension L of K, there exists an irreducible admissible representation BCL/K(1r) of GL(2, L), the base change of ir to L [9, Section 2]. This representation of GL(2, L) enters into the expression for the character. The main result of this paper is the following formula. Recall that conjugacy classes of nonsplit Cartan subgroups are parameterized by quadratic separable field extensions L of K, via embeddings of L* in GL(2, K).

The Jacquet–Langlands Correspondence via Twisted Descent

2015-10-29
Dihua Jiang , Baiying Liu , Bin Xu +1 more
The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained 
 The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 [13]. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972 [30]. In this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry [10] to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence for |${\mathrm {PGL}}(2)$| via a new explicit construction.
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Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems

2000-07-01
E. Ait Dads , Khalil Ezzinbi

Existence of positive almost periodic solutions of functional equations via hilbert's protective metric

1996-03-01
Khalil Ezzinbi , M.A. Hachimi

Pseudo Almost Periodic Solutions of Some Differential Equations, II

1995-06-01
C.Y. Zhang

Positive almost periodic solutions of a state-dependent delay nonlinear integral equation

1993-06-01
Ricardo TorrejĂłn

On Tunnell’s formula for characters of $GL(2)$

1993-01-01
Hiroshi Saitƍ

On Ambrosetti–Malchiodi–Ni Conjecture for General Hypersurfaces

2011-10-25
Liping Wang , Juncheng Wei , Jun Yang
Abstract We consider the nonlinear problem where p > 1, Ï” is a small parameter and V is a uniformly positive, smooth potential. Assume that 𝒩 ⊂ ℝ n is 
 Abstract We consider the nonlinear problem where p > 1, Ï” is a small parameter and V is a uniformly positive, smooth potential. Assume that 𝒩 ⊂ ℝ n is a smooth closed, stationary and non-degenerate hypersurface relative to the functional with . We prove the existence of solutions , at least for some sequence {ϔℓ}ℓ, which concentrate along smooth surfaces Γϔ close to 𝒩. This result confirms the validity of the conjecture of Ambrosetti et al. in [Citation2] for concentration of Schrödinger equation on general hypersurfaces. Keywords: Ambrosetti–Malchiodi–Ni conjectureConcentrationInfinite-dimensional reductionNonlinear Schrödinger equationMathematics Subject Classification: 35J2535J2035B3335B40 Acknowledgments The first author is supported by NSFC 10901053 and ECNU 78210052. The second author is supported by an Earmarked Grant from RGC of Hong Kong. The third author is supported by the foundations NSFC (No. 10901108), NSF of Guangdong (No. 10451806001004770) and 801-000012 of SZU R/F. Part of this work was done when the third author visited the Chern Institute of Mathematics, Nankai University; he is very grateful to the institution for the kind hospitality. We thank M. del Pino and M. Kowalczyk for useful conversations. Last but not the least, we are very grateful to the referees for pointing out a gap in the first version of the manuscript.

Positive solutions of nonlinear integral equations arising in infectious diseases

1988-08-01
Dajun Guo , V. Lakshmikantham

On a singularly perturbed equation with neumann boundary condition

1998-01-01
Yanyan Li
(1998). On a singularly perturbed equation with neumann boundary condition. Communications in Partial Differential Equations: Vol. 23, No. 3-4, pp. 487-545. (1998). On a singularly perturbed equation with neumann boundary condition. Communications in Partial Differential Equations: Vol. 23, No. 3-4, pp. 487-545.

Sturm–liouville and dirac operators

1992-05-01
Levitan , I. Sargsjan
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A Product of Tensor Product L-functions of Quasi-split Classical Groups of Hermitian Type

2014-02-04
Dihua Jiang , Lei Zhang
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Almost-Periodic Functions and Functional Equations

1971-01-01
Luigi Amerio , Giovanni Prouse

On the existence of some new positive interior spike solutions to a semilinear Neumann problem

2009-08-06
Teresa D’Aprile , Angela Pistoia

On the nonvanishing of the central value of the Rankin-Selberg 𝐿-functions

2004-04-01
David Ginzburg , Dihua Jiang , Stephen Rallis
We characterize the nonvanishing of the central value of the Rankin-Selberg <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions in terms of periods of Fourier-Jacobi 
 We characterize the nonvanishing of the central value of the Rankin-Selberg <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions in terms of periods of Fourier-Jacobi type. This characterization is based on the Langlands philosophy on functoriality and on applications of invariant theory to automorphic representations. The result is the symplectic analog of the Gross-Prasad conjecture.

On the cuspidal spectrum for finite volume symmetric spaces

1982-01-01
Harold Donnelly
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On the shape of least‐energy solutions to a semilinear Neumann problem

1991-09-01
Wei‐Ming Ni , Izumi Takagi

EXISTENCE AND STABILITY OF THREE-DIMENSIONAL BOUNDARY-INTERIOR LAYERS FOR THE ALLEN-CAHN EQUATION

2005-01-09
Kunimochi Sakamoto
A minimal surface intersecting the boundary of a smooth bounded domain $\subset\mathbb{R}^3$, when it is {\em non-degenerate}, gives rise to a family of transition layer solutions of the Allen-Cahn equation. 
 A minimal surface intersecting the boundary of a smooth bounded domain $\subset\mathbb{R}^3$, when it is {\em non-degenerate}, gives rise to a family of transition layer solutions of the Allen-Cahn equation. The stability properties of the transition layer solution are determined by the eigenvalues of the Jacobi operator on the minimal surface with Robin type boundary conditions which encode the geometric information of the domain boundary.

Locating the peaks of least-energy solutions to a semilinear Neumann problem

1993-05-01
Wei-Ming Ni , Izumi Takagi

On spikes concentrating on line-segments to a semilinear Neumann problem

2011-05-23
Weiwei Ao , Monica Musso , Juncheng Wei

Boundary concentration phenomena for a singularly perturbed elliptic problem

2002-09-27
Andrea Malchiodi , Marcelo Montenegro
Abstract We exhibit new concentration phenomena for the equation − Δ 2 Δ u + u = u p in a smooth bounded domain Ω ⊆ ℝ 2 and with 
 Abstract We exhibit new concentration phenomena for the equation − Δ 2 Δ u + u = u p in a smooth bounded domain Ω ⊆ ℝ 2 and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter Δ is converging to 0. For a suitable sequence Δ n → 0 we prove the existence of positive solutions u n concentrating at the whole boundary of Ω or at some component. © 2002 Wiley Periodicals, Inc.

L’endoscopie tordue n’est pas si tordue

2008-01-01
J. L. Waldspurger
Inroduction La conjecture de transfert Analyse harmonique Classes de conjugaison stable et correspondances endoscopiques Le cas non ramifie Cas non ramifie: les preuves Preliminaires cohomologiques Definition des facteurs de transfert 
 Inroduction La conjecture de transfert Analyse harmonique Classes de conjugaison stable et correspondances endoscopiques Le cas non ramifie Cas non ramifie: les preuves Preliminaires cohomologiques Definition des facteurs de transfert Normalisation du facteur de transfert dans le cas non ramifie Rapport de facteurs de transfert Egalite de facteurs de transfert Reduction a un sous-groupe de Levi Reduction a une situation non ramifiee Reduction au cas quasi-simple Le cas $\theta=1$ Le cas: $G^*$ de type $A_{n-1}$ Le cas: $G^*$ de $D_{4}$ et $\theta$ d'ordre $3$ Le cas: $G^*$ de type $D_{n}$ et $\theta$ d'ordre $2$ Le cas: $G\*$ de type $E_{6}$ et $\theta$ d'ordre $2$ Appendice A: sections d'extensions Appendice B: l'exponentielle Bibliographie.

On the Boundary Spike Layer Solutions to a Singularly Perturbed Neumann Problem

1997-02-01
Juncheng Wei

Stationary solutions for the Cahn–Hilliard equation

1998-08-01
Juncheng Wei , Matthias Winter
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