Pub Date : 2025-01-23DOI: 10.1016/j.indag.2025.01.002
Dean Wardell
A finitely generated group is called strongly scale-invariant if there exists an injective homomorphism such that is a finite index subgroup of and such that is finite. Nekrashevych and Pete conjectured that all strongly scale-invariant groups are virtually nilpotent, after disproving a stronger conjecture by Benjamini.
This conjecture is known to be true in some situations. Deré proved it for virtually polycyclic groups. In this paper, we provide an elementary proof for those polycyclic groups that can be written as a semi-direct product .
{"title":"An elementary proof of the Benjamini–Nekrashevych–Pete conjecture for the semi-direct products Zn⋊Z","authors":"Dean Wardell","doi":"10.1016/j.indag.2025.01.002","DOIUrl":"10.1016/j.indag.2025.01.002","url":null,"abstract":"<div><div>A finitely generated group <span><math><mi>G</mi></math></span> is called strongly scale-invariant if there exists an injective homomorphism <span><math><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is a finite index subgroup of <span><math><mi>G</mi></math></span> and such that <span><math><mrow><msub><mrow><mo>∩</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is finite. Nekrashevych and Pete conjectured that all strongly scale-invariant groups are virtually nilpotent, after disproving a stronger conjecture by Benjamini.</div><div>This conjecture is known to be true in some situations. Deré proved it for virtually polycyclic groups. In this paper, we provide an elementary proof for those polycyclic groups that can be written as a semi-direct product <span><math><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⋊</mo><mi>Z</mi></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1125-1132"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-03DOI: 10.1016/j.indag.2024.12.005
Martin Mathieu , Francois Schulz
We establish Jafarian’s 2009 conjecture that every additive spectrum preserving mapping from a von Neumann algebra onto a semisimple Banach algebra is a Jordan isomorphism.
{"title":"Additive spectrum preserving mappings fromvon Neumann algebras","authors":"Martin Mathieu , Francois Schulz","doi":"10.1016/j.indag.2024.12.005","DOIUrl":"10.1016/j.indag.2024.12.005","url":null,"abstract":"<div><div>We establish Jafarian’s 2009 conjecture that every additive spectrum preserving mapping from a von Neumann algebra onto a semisimple Banach algebra is a Jordan isomorphism.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 4","pages":"Pages 1112-1123"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.03.014
Adam Korányi
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.
{"title":"A simplified approach to the holomorphic discrete series","authors":"Adam Korányi","doi":"10.1016/j.indag.2024.03.014","DOIUrl":"10.1016/j.indag.2024.03.014","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 29-41"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.03.006
V.F. Molchanov
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.
{"title":"Berezin quantization and representation theory","authors":"V.F. Molchanov","doi":"10.1016/j.indag.2024.03.006","DOIUrl":"10.1016/j.indag.2024.03.006","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 14-28"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.04.010
Wolter Groenevelt , Joop Vermeulen
We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group . Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.
{"title":"Multivariate Meixner polynomials related to holomorphic discrete series representations of SU(1,d)","authors":"Wolter Groenevelt , Joop Vermeulen","doi":"10.1016/j.indag.2024.04.010","DOIUrl":"10.1016/j.indag.2024.04.010","url":null,"abstract":"<div><div>We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group <span><math><mrow><mi>SU</mi><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 171-187"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141033600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.05.008
Toshihisa Kubo , Bent Ørsted
We classify and construct -intertwining differential operators from a line bundle to a vector bundle over the real projective space by the F-method. This generalizes a classical result of Bol for . Further, we classify the -type formulas for the kernel and image of . The standardness of the homomorphisms corresponding to the differential operators between generalized Verma modules is also discussed.
我们用 F 方法对实射空间上从线束到向量束的-交织微分算子进行了分类和构造。这概括了波尔关于.的经典结果。此外，我们还讨论了广义 Verma 模块之间微分算子对应的同态的标准性。
{"title":"On the intertwining differential operators from a line bundle to a vector bundle over the real projective space","authors":"Toshihisa Kubo , Bent Ørsted","doi":"10.1016/j.indag.2024.05.008","DOIUrl":"10.1016/j.indag.2024.05.008","url":null,"abstract":"<div><div>We classify and construct <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span><span>-intertwining differential operators </span><span><math><mi>D</mi></math></span> from a line bundle to a vector bundle over the real projective space <span><math><mrow><mi>R</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> by the F-method. This generalizes a classical result of Bol for <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. Further, we classify the <span><math><mi>K</mi></math></span>-type formulas for the kernel <span><math><mrow><mi>Ker</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and image <span><math><mrow><mi>Im</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>D</mi></math></span><span>. The standardness of the homomorphisms </span><span><math><mi>φ</mi></math></span> corresponding to the differential operators <span><math><mi>D</mi></math></span> between generalized Verma modules is also discussed.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 270-301"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.04.002
Jan Frahm , Karl-Hermann Neeb , Gestur Ólafsson
This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups acting on a non-compactly causal symmetric space , every irreducible unitary representation of can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over . In the present paper we discuss this procedure for the connected Lorentz group acting on de Sitter space . We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain in the complexified group , which for specializes to the complex cut plane . A number of special situations is discussed specifically: (a) The case , which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case , where we obtain rather explicit formulas for the matrix-valued spherical functions.
{"title":"Realization of unitary representations of the Lorentz group on de Sitter space","authors":"Jan Frahm , Karl-Hermann Neeb , Gestur Ólafsson","doi":"10.1016/j.indag.2024.04.002","DOIUrl":"10.1016/j.indag.2024.04.002","url":null,"abstract":"<div><div>This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups <span><math><mi>G</mi></math></span> acting on a non-compactly causal symmetric space <span><math><mrow><mi>M</mi><mo>=</mo><mi>G</mi><mo>/</mo><mi>H</mi></mrow></math></span>, every irreducible unitary representation of <span><math><mi>G</mi></math></span> can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over <span><math><mi>M</mi></math></span>. In the present paper we discuss this procedure for the connected Lorentz group <span><math><mrow><mi>G</mi><mo>=</mo><msub><mrow><mi>SO</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>d</mi></mrow></msub><msub><mrow><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow><mrow><mi>e</mi></mrow></msub></mrow></math></span> acting on de Sitter space <span><math><mrow><mi>M</mi><mo>=</mo><msup><mrow><mi>dS</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>ℂ</mi></mrow><mrow><mi>cut</mi></mrow></msubsup></math></span> in the complexified group <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>ℂ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>SO</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>, which for <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span> specializes to the complex cut plane <span><math><mrow><mi>ℂ</mi><mo>∖</mo><mrow><mo>(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn><mo>]</mo></mrow></mrow></math></span>. A number of special situations is discussed specifically: (a) The case <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span>, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for <span><math><mrow><mi>d</mi><mo>></mo><mn>2</mn></mrow></math></span> is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case <span><math><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math></span>, where we obtain rather explicit formulas for the matrix-valued spherical functions.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 61-113"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.04.009
Gert Heckman
In 2004 Gerrit retired as professor of mathematics from Leiden University. In the evening there was a nice dinner party on the occasion with several speeches. As first Ph.D. student of Gerrit I was also asked to say a few words. The main point I made was that Gerrit had been for me the right man at the right time. At the end of the evening Gerrit was the last speaker. He thanked all the speakers one by one for their nice words. To me he said that I had exaggerated a little and as an independent student had found my own way. In this note I will discuss my Ph.D. period with Gerrit and maybe it will become clear why we both said what we said then.
{"title":"Becoming a mathematician","authors":"Gert Heckman","doi":"10.1016/j.indag.2024.04.009","DOIUrl":"10.1016/j.indag.2024.04.009","url":null,"abstract":"<div><div>In 2004 Gerrit retired as professor of mathematics from Leiden University. In the evening there was a nice dinner party on the occasion with several speeches. As first Ph.D. student of Gerrit I was also asked to say a few words. The main point I made was that Gerrit had been for me the right man at the right time. At the end of the evening Gerrit was the last speaker. He thanked all the speakers one by one for their nice words. To me he said that I had exaggerated a little and as an independent student had found my own way. In this note I will discuss my Ph.D. period with Gerrit and maybe it will become clear why we both said what we said then.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 4-13"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141037961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.05.004
Dominik Brennecken, Margit Rösler
We study the asymptotic behaviour of Bessel functions associated to root systems of type and type with positive multiplicities as the rank tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of -ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.
我们研究了当阶数趋于无穷大时，与正乘数类型和类型根系统相关的贝塞尔函数的渐近行为。在这两种情况下，我们都描述了可能的极限函数以及存在这些极限的光谱参数的 Vershik-Kerov 类型序列。在类型情况下，这为阿西奥蒂斯（Assiotis）和纳吉努德尔（Najnudel）在随机矩阵理论中的-集合背景下的最新结果提供了一种新的和非常自然的方法。这些结果概括了关于用有限维赫米提矩阵空间的球形函数逼近无限维赫米提矩阵空间的正有限奥尔森斯基球形函数（具有相关无限单元群的作用）的已知事实。在 B 型情况下，我们的结果包括与非紧密格拉斯曼的 Cartan 运动群相关的球函数在秩达到无穷大时的渐近结果，以及相关归纳极限的 Olshanski 球函数的分类。
{"title":"Limits of Bessel functions for root systems as the rank tends to infinity","authors":"Dominik Brennecken, Margit Rösler","doi":"10.1016/j.indag.2024.05.004","DOIUrl":"10.1016/j.indag.2024.05.004","url":null,"abstract":"<div><div>We study the asymptotic behaviour of Bessel functions associated to root systems of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with positive multiplicities as the rank <span><math><mi>n</mi></math></span> tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type <span><math><mi>A</mi></math></span> case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of <span><math><mi>β</mi></math></span>-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over <span><math><mrow><mi>F</mi><mo>=</mo><mi>R</mi><mo>,</mo><mi>ℂ</mi><mo>,</mo><mi>H</mi></mrow></math></span> (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 245-269"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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