{"title":"On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection","authors":"Alexandre Afgoustidis","doi":"10.4310/cjm.2021.v9.n3.a1","DOIUrl":null,"url":null,"abstract":"George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between \"most\" irreducible (tempered) representations of $G$ and \"most\" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2015-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2021.v9.n3.a1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 7
Abstract
George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.
George Mackey在1975年提出了非紧约李群$G$的不可约酉表示与它的Cartan运动群$G_0$ $-$ G$的极大紧子群与向量空间的半直积的不可约酉表示之间存在类比。他推测在$G$的“最”不可约(调质)表示和$G_0$的“最”不可约(酉)表示之间存在一种自然的一对一对应关系。我们在这里描述了两个群的调和对偶之间的简单和自然的双射,以及可容许对偶之间一对一对应的扩展。