{"title":"A simplified approach to the holomorphic discrete series","authors":"Adam Korányi","doi":"10.1016/j.indag.2024.03.014","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.014","url":null,"abstract":"Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantum superintegrable spin systems on graph connections","authors":"Nicolai Reshetikhin, Jasper Stokman","doi":"10.1016/j.indag.2024.03.008","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.008","url":null,"abstract":"In this paper we construct certain quantum spin systems on moduli spaces of -connections on a connected oriented finite graph, with a simply connected compact Lie group. We construct joint eigenfunctions of the commuting quantum Hamiltonians in terms of local invariant tensors. We determine sufficient conditions ensuring superintegrability of the quantum spin system using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra and Lepowsky & McCollum. The resulting class of quantum superintegrable spin systems includes the quantum periodic and open spin Calogero–Moser spin chains as special cases. In the periodic case the description of the joint eigenfunctions in terms of local invariant tensors are multipoint generalized trace functions, in the open case multipoint spherical functions on compact symmetric spaces.","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symplectic complexity of reductive group actions","authors":"Avraham Aizenbud, Dmitry Gourevitch","doi":"10.1016/j.indag.2024.03.010","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.010","url":null,"abstract":"Let a complex algebraic reductive group act on a complex algebraic manifold . For a -invariant subvariety of the nilpotent cone we define a notion of -symplectic complexity of . This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of -complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generating operators of symmetry breaking — From discrete to continuous","authors":"Toshiyuki Kobayashi","doi":"10.1016/j.indag.2024.03.007","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.007","url":null,"abstract":"Based on the “generating operator” of the Rankin–Cohen bracket introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Berezin quantization and representation theory","authors":"V.F. Molchanov","doi":"10.1016/j.indag.2024.03.006","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.006","url":null,"abstract":"We present an approach to Berezin quantization (a variant of quantization in the spirit of Berezin) on para-Hermitian symmetric spaces using the notion of an “overgroup”. This approach gives covariant and contravariant symbols and the Berezin transform in a natural and transparent way.","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symmetric pairs and branching laws","authors":"Paul-Émile Paradan","doi":"10.1016/j.indag.2024.03.009","DOIUrl":"https://doi.org/10.1016/j.indag.2024.03.009","url":null,"abstract":"Let be a compact connected Lie group and let be a subgroup fixed by an involution. A classical result assures that the -action on the flag variety of admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair that is parametrized by .","PeriodicalId":501252,"journal":{"name":"Indagationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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