{"title":"与 Sp(1, n) 主数列表示相关的傅立叶-泊松变换","authors":"Xingya Fan, Jianxun He, Xiaoke Jia","doi":"10.1007/s00006-024-01330-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(X=Sp(1,n)/Sp(n)\\)</span> be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where <i>n</i> is greater than or equal to 1. The Fürstenberg boundary of <i>X</i> is denoted as <span>\\(\\Sigma \\)</span>. In this paper, we focus on the Plancherel formula on <i>X</i> associated with the Poisson transform of vector-valued <span>\\(L^2\\)</span>-space on <span>\\(\\Sigma \\)</span>. Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of <i>Sp</i>(1, <i>n</i>) on <span>\\(L^2(X)\\)</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)\",\"authors\":\"Xingya Fan, Jianxun He, Xiaoke Jia\",\"doi\":\"10.1007/s00006-024-01330-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\(X=Sp(1,n)/Sp(n)\\\\)</span> be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where <i>n</i> is greater than or equal to 1. The Fürstenberg boundary of <i>X</i> is denoted as <span>\\\\(\\\\Sigma \\\\)</span>. In this paper, we focus on the Plancherel formula on <i>X</i> associated with the Poisson transform of vector-valued <span>\\\\(L^2\\\\)</span>-space on <span>\\\\(\\\\Sigma \\\\)</span>. Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of <i>Sp</i>(1, <i>n</i>) on <span>\\\\(L^2(X)\\\\)</span>.</p></div>\",\"PeriodicalId\":7330,\"journal\":{\"name\":\"Advances in Applied Clifford Algebras\",\"volume\":\"34 3\",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-05-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Applied Clifford Algebras\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00006-024-01330-1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Clifford Algebras","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00006-024-01330-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)
Let \(X=Sp(1,n)/Sp(n)\) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as \(\Sigma \). In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued \(L^2\)-space on \(\Sigma \). Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of Sp(1, n) on \(L^2(X)\).
期刊介绍:
Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.