{"title":"还原对偶(U[式略],U[式略])的对称破缺算子","authors":"M. McKee , A. Pasquale , T. Przebinda","doi":"10.1016/j.indag.2024.06.004","DOIUrl":null,"url":null,"abstract":"<div><div>We consider the dual pair <span><math><mrow><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>)</mo></mrow></mrow></math></span> in the symplectic group <span><math><mrow><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. Fix a Weil representation of the metaplectic group <span><math><mrow><msub><mrow><mover><mrow><mi>Sp</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and <span><math><msup><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></math></span> be the preimages of <span><math><mi>G</mi></math></span> and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> in <span><math><mrow><msub><mrow><mover><mrow><mi>Sp</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, and let <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> be a genuine irreducible representation of <span><math><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>×</mo><msup><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. We study the Weyl symbol <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span><span> of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with </span><span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its <span><math><mi>Π</mi></math></span>-isotypic component and also with the orthogonal projection onto its <span><math><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>-isotypic component. Hence <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> can be computed in two different ways, one using <span><math><mi>Π</mi></math></span> and the other using <span><math><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. By matching the results, we recover Weyl’s theorem stating that <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> occurring in Howe’s correspondence.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 413-449"},"PeriodicalIF":0.5000,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Symmetry breaking operators for the reductive dual pair (Ul,Ul′)\",\"authors\":\"M. McKee , A. Pasquale , T. Przebinda\",\"doi\":\"10.1016/j.indag.2024.06.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider the dual pair <span><math><mrow><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>)</mo></mrow></mrow></math></span> in the symplectic group <span><math><mrow><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. Fix a Weil representation of the metaplectic group <span><math><mrow><msub><mrow><mover><mrow><mi>Sp</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and <span><math><msup><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></math></span> be the preimages of <span><math><mi>G</mi></math></span> and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> in <span><math><mrow><msub><mrow><mover><mrow><mi>Sp</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, and let <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> be a genuine irreducible representation of <span><math><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>×</mo><msup><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. We study the Weyl symbol <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span><span> of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with </span><span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its <span><math><mi>Π</mi></math></span>-isotypic component and also with the orthogonal projection onto its <span><math><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>-isotypic component. Hence <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> can be computed in two different ways, one using <span><math><mi>Π</mi></math></span> and the other using <span><math><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. By matching the results, we recover Weyl’s theorem stating that <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations <span><math><mrow><mi>Π</mi><mo>⊗</mo><msup><mrow><mi>Π</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> occurring in Howe’s correspondence.</div></div>\",\"PeriodicalId\":56126,\"journal\":{\"name\":\"Indagationes Mathematicae-New Series\",\"volume\":\"36 2\",\"pages\":\"Pages 413-449\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-06-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae-New Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000685\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000685","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们考虑交映组中的对偶 .固定一个元折射群的 Weil 表示.让 和 成为 和 在 的前像,让 成为 的真正不可还原表示. 我们研究 Weil 表示与......交织在一起的对称破缺算子(SBO)的韦尔符号(唯一的,直到一个可能为零的常数)。这个 SBO 与 Weil 表示的空间对其-异型分量的正交投影以及对其-异型分量的正交投影重合。因此,可以用两种不同的方法计算,一种是使用...,另一种是使用...。通过匹配结果,我们恢复了韦尔定理,即在韦尔表示中出现的乘数至多为一,我们还恢复了豪氏对应关系中出现的表示的完整列表。
Symmetry breaking operators for the reductive dual pair (Ul,Ul′)
We consider the dual pair in the symplectic group . Fix a Weil representation of the metaplectic group . Let and be the preimages of and in , and let be a genuine irreducible representation of . We study the Weyl symbol of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with . This SBO coincides with the orthogonal projection of the space of the Weil representation onto its -isotypic component and also with the orthogonal projection onto its -isotypic component. Hence can be computed in two different ways, one using and the other using . By matching the results, we recover Weyl’s theorem stating that occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations occurring in Howe’s correspondence.
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.