{"title":"Heisenberg群扩展的Hecke-Baxter算子","authors":"A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin","doi":"10.1007/s11005-025-01971-8","DOIUrl":null,"url":null,"abstract":"<div><p>The <span>\\(GL_{\\ell +1}(\\mathbb {R})\\)</span> Hecke-Baxter operator was introduced as an element of the <span>\\(O_{\\ell +1}\\)</span>-spherical Hecke algebra associated with the Gelfand pair <span>\\(O_{\\ell +1}\\subset GL_{\\ell +1}(\\mathbb {R})\\)</span>. It was specified by the property to act on an <span>\\(O_{\\ell +1}\\)</span>-fixed vector in a <span>\\(GL_{\\ell +1}(\\mathbb {R})\\)</span>-principal series representation via multiplication by the local Archimedean <i>L</i>-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group <span>\\(GL_{\\ell +1}(\\mathbb {R})\\times GL_{\\ell +1}(\\mathbb {R})\\)</span> by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group <span>\\(Sp_{2\\ell +2}(\\mathbb {R})\\times Sp_{2\\ell +2}(\\mathbb {R})\\)</span> by a Heisenberg Lie group.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 4","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Hecke-Baxter operators via Heisenberg group extensions\",\"authors\":\"A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin\",\"doi\":\"10.1007/s11005-025-01971-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The <span>\\\\(GL_{\\\\ell +1}(\\\\mathbb {R})\\\\)</span> Hecke-Baxter operator was introduced as an element of the <span>\\\\(O_{\\\\ell +1}\\\\)</span>-spherical Hecke algebra associated with the Gelfand pair <span>\\\\(O_{\\\\ell +1}\\\\subset GL_{\\\\ell +1}(\\\\mathbb {R})\\\\)</span>. It was specified by the property to act on an <span>\\\\(O_{\\\\ell +1}\\\\)</span>-fixed vector in a <span>\\\\(GL_{\\\\ell +1}(\\\\mathbb {R})\\\\)</span>-principal series representation via multiplication by the local Archimedean <i>L</i>-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group <span>\\\\(GL_{\\\\ell +1}(\\\\mathbb {R})\\\\times GL_{\\\\ell +1}(\\\\mathbb {R})\\\\)</span> by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group <span>\\\\(Sp_{2\\\\ell +2}(\\\\mathbb {R})\\\\times Sp_{2\\\\ell +2}(\\\\mathbb {R})\\\\)</span> by a Heisenberg Lie group.</p></div>\",\"PeriodicalId\":685,\"journal\":{\"name\":\"Letters in Mathematical Physics\",\"volume\":\"115 4\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2025-07-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Letters in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11005-025-01971-8\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-025-01971-8","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
The Hecke-Baxter operators via Heisenberg group extensions
The \(GL_{\ell +1}(\mathbb {R})\) Hecke-Baxter operator was introduced as an element of the \(O_{\ell +1}\)-spherical Hecke algebra associated with the Gelfand pair \(O_{\ell +1}\subset GL_{\ell +1}(\mathbb {R})\). It was specified by the property to act on an \(O_{\ell +1}\)-fixed vector in a \(GL_{\ell +1}(\mathbb {R})\)-principal series representation via multiplication by the local Archimedean L-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group \(GL_{\ell +1}(\mathbb {R})\times GL_{\ell +1}(\mathbb {R})\) by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group \(Sp_{2\ell +2}(\mathbb {R})\times Sp_{2\ell +2}(\mathbb {R})\) by a Heisenberg Lie group.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.