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Transfer of Highest Weight Modules and Small Unipotent Representations 最高权重模块和小单能表示的转移
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-3237-4
Hai An He, Jing Song Huang, Kayue Daniel Wong
{"title":"Transfer of Highest Weight Modules and Small Unipotent Representations","authors":"Hai An He, Jing Song Huang, Kayue Daniel Wong","doi":"10.1007/s10114-024-3237-4","DOIUrl":"https://doi.org/10.1007/s10114-024-3237-4","url":null,"abstract":"<p>We study the transfer between small special unipotent representations for all equal rank real forms of type <i>E</i><sub>6</sub> and <i>E</i><sub>7</sub>. As a consequence, one can verify these modules are unitarity using the results of Wallach and Zhu. Moreover, the <i>K</i>-spectra of these modules can be obtained explicitly.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries 具有双面片断线性边界的布朗运动的首过密度
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-1090-0
Zhen Yu, Mao Zai Tian
{"title":"First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries","authors":"Zhen Yu, Mao Zai Tian","doi":"10.1007/s10114-024-1090-0","DOIUrl":"https://doi.org/10.1007/s10114-024-1090-0","url":null,"abstract":"<p>The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Wave Front Set Correspondence for Dual Pairs with One Member Compact 有一个成员的紧凑型双对波前集合对应关系
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-1424-y
Mark McKee, Angela Pasquale, Tomasz Przebinda
{"title":"The Wave Front Set Correspondence for Dual Pairs with One Member Compact","authors":"Mark McKee, Angela Pasquale, Tomasz Przebinda","doi":"10.1007/s10114-024-1424-y","DOIUrl":"https://doi.org/10.1007/s10114-024-1424-y","url":null,"abstract":"<p>Let W be a real symplectic space and (G, G′) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let <span>(widetilde {rm{G}})</span> be the preimage of G in the metaplectic group <span>(widetilde {{rm{Sp}}}({rm{W}}))</span>. Given an irreducible unitary representation Π of <span>(widetilde {rm{G}})</span> that occurs in the restriction of the Weil representation to <span>(widetilde {rm{G}})</span>, let Θ<sub>Π</sub> denote its character. We prove that, for a suitable embedding <i>T</i> of <span>(widetilde {{rm{Sp}}}({rm{W}}))</span> in the space of tempered distributions on W, the distribution <i>T</i>(Θ̌<sub>Π</sub>) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π′, the representation of <span>(widetilde {{G^prime}})</span> dual to Π, by elementary means.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solution of the Center Problem for a Class of Polynomial Differential Systems 一类多项式微分方程系统的中心问题求解
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-0578-y
Chang Jian Liu, Jaume Llibre, Rafael Ramírez, Valentín Ramírez
{"title":"Solution of the Center Problem for a Class of Polynomial Differential Systems","authors":"Chang Jian Liu, Jaume Llibre, Rafael Ramírez, Valentín Ramírez","doi":"10.1007/s10114-024-0578-y","DOIUrl":"https://doi.org/10.1007/s10114-024-0578-y","url":null,"abstract":"<p>Consider the polynomial differential system of degree <i>m</i> of the form </p><span>$$eqalign{&amp;dot{x}=-y(1+mu(a_{2}x-a_{1}y))+x(nu(a_{1}x+a_{2}y)+Omega_{m-1}(x,y)),cr &amp;dot{y}=x(1+mu(a_{2}x-a_{1}y))+y(nu(a_{1}x+a_{2}y)+Omega_{m-1}(x,y)),}$$</span><p> where <i>μ</i> and <i>ν</i> are real numbers such that <span>((mu^{2}+nu^{2})(mu+nu(m-2))(a_{1}^{2}+a_{2}^{2})ne 0,m &gt; 2)</span> and Ω<sub><i>m</i>−1</sub>(<i>x</i>,<i>y</i>) is a homogenous polynomial of degree <i>m</i> − 1. A conjecture, stated in <i>J. Differential Equations</i> 2019, suggests that when <i>ν</i> = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (<i>x</i>,<i>y</i>) → (<i>X</i>,<i>Y</i>) the system is invariant under the transformation (<i>X</i>,<i>Y</i>,<i>t</i>) → (−<i>X</i>,<i>Y</i>, −<i>t</i>). For every degree <i>m</i> we prove the extension of this conjecture to any value of <i>ν</i> except for a finite set of values of <i>μ</i>.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras 古典列阵的格尔芬-基里洛夫维度和标量广义维尔马模块的可重复性
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-2676-2
Zhan Qiang Bai, Jing Jiang
{"title":"Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras","authors":"Zhan Qiang Bai, Jing Jiang","doi":"10.1007/s10114-024-2676-2","DOIUrl":"https://doi.org/10.1007/s10114-024-2676-2","url":null,"abstract":"<p>Let <span>(mathfrak{g})</span> be a classical complex simple Lie algebra and <span>(mathfrak{q})</span> be a parabolic subalgebra. Let <i>M</i> be a generalized Verma module induced from a one dimensional representation of <span>(mathfrak{q})</span>. Such <i>M</i> is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand–Kirillov dimension of the corresponding highest weight modules.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Irreducible Representations of GLn(ℂ) of Minimal Gelfand–Kirillov Dimension 最小格尔芬-基里洛夫维度的 GLn(ℂ) 不可还原表征
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-3207-x
Zhan Qiang Bai, Yang Yang Chen, Dong Wen Liu, Bin Yong Sun
{"title":"Irreducible Representations of GLn(ℂ) of Minimal Gelfand–Kirillov Dimension","authors":"Zhan Qiang Bai, Yang Yang Chen, Dong Wen Liu, Bin Yong Sun","doi":"10.1007/s10114-024-3207-x","DOIUrl":"https://doi.org/10.1007/s10114-024-3207-x","url":null,"abstract":"<p>In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of <i>G</i> = GL<sub><i>n</i></sub>(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of <i>G</i> of type (<i>n</i> − 1,1). We give the transition matrix between the two bases for the corresponding coherent families.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140037039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some Rotundities of Orlicz–Lorentz Spaces 奥尔利奇-洛伦兹空间的一些轮回性
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-15 DOI: 10.1007/s10114-024-2551-1
Wan Zhong Gong, Peng Wang
{"title":"Some Rotundities of Orlicz–Lorentz Spaces","authors":"Wan Zhong Gong, Peng Wang","doi":"10.1007/s10114-024-2551-1","DOIUrl":"https://doi.org/10.1007/s10114-024-2551-1","url":null,"abstract":"<h3>Abstract</h3> <p>K-UR, K-LUR and K-R are the generalizations of UR, LUR and R respectively, which are of great significance in Banach space theory. While in Orlicz–Lorentz function space <span> <span>(Lambda_{varphi,omega}^{circ}[0,gamma))</span> </span> equipped with the Orlicz norm, the research methods of K-UR, K-LUR and K-R are much more complicated than those of UR, LUR and R. In this paper we obtain some criteria of K-UR, K-LUR and K-R of <span> <span>(Lambda_{varphi,omega}^{circ}[0,gamma))</span> </span> by means of the norm of dual space and <em>H</em><sub><em>μ</em></sub> property of <span> <span>(Lambda_{varphi,omega}^{circ}[0,gamma))</span> </span>.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Big Theta Equals Small Theta Generically 大 Theta 一般等于小 Theta
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-03-01 DOI: 10.1007/s10114-024-3236-5
Rui Chen, Jia Liang Zou
{"title":"Big Theta Equals Small Theta Generically","authors":"Rui Chen, Jia Liang Zou","doi":"10.1007/s10114-024-3236-5","DOIUrl":"https://doi.org/10.1007/s10114-024-3236-5","url":null,"abstract":"<h3>Abstract</h3> <p>In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140037306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quaternionic Monge–Ampère Measure on Pluripolar Set 多极集上的四元蒙日-安培测度
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-02-02 DOI: 10.1007/s10114-024-2227-x
Hichame Amal, Saïd Asserda, Fadoua Boukhari
{"title":"Quaternionic Monge–Ampère Measure on Pluripolar Set","authors":"Hichame Amal, Saïd Asserda, Fadoua Boukhari","doi":"10.1007/s10114-024-2227-x","DOIUrl":"https://doi.org/10.1007/s10114-024-2227-x","url":null,"abstract":"<p>In this paper, we prove that in a hyperconvex domain Ω in ℍ<sup><i>n</i></sup>, if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class <span>({cal E}(Omega ))</span>.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ungraded Matrix Factorizations as Mirrors of Non-orientable Lagrangians 无等级矩阵因式分解作为不可定向拉格朗日的镜像
IF 0.7 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-01-15 DOI: 10.1007/s10114-024-2268-1
Lino Amorim, Cheol-Hyun Cho
{"title":"Ungraded Matrix Factorizations as Mirrors of Non-orientable Lagrangians","authors":"Lino Amorim, Cheol-Hyun Cho","doi":"10.1007/s10114-024-2268-1","DOIUrl":"https://doi.org/10.1007/s10114-024-2268-1","url":null,"abstract":"<p>We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial <i>W</i>, with coefficients in a field of characteristic 2, is a square matrix <i>Q</i> of polynomial entries satisfying <i>Q</i><sup>2</sup> = <i>W</i> · Id. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold ℝ<i>P</i><sup>2</sup> ⊂ ℂ<i>P</i><sup>2</sup> and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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