Indagationes Mathematicae-New Series最新文献

{"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}

{"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}

{"title":"On the intertwining differential operators from a line bundle to a vector bundle over the real projective space","authors":"Toshihisa Kubo ,&nbsp;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}

{"title":"Multivariate Meixner polynomials related to holomorphic discrete series representations of SU(1,d)","authors":"Wolter Groenevelt ,&nbsp;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}

{"title":"Realization of unitary representations of the Lorentz group on de Sitter space","authors":"Jan Frahm ,&nbsp;Karl-Hermann Neeb ,&nbsp;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>&gt;</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}

{"title":"Gerrit van Dijk (1939–2022)","authors":"Marcel de Jeu,&nbsp;Erik Koelink,&nbsp;Eric Opdam,&nbsp;Michael Pevzner","doi":"10.1016/j.indag.2024.09.004","DOIUrl":"10.1016/j.indag.2024.09.004","url":null,"abstract":"","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 1-3"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145652","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}

{"title":"Limits of Bessel functions for root systems as the rank tends to infinity","authors":"Dominik Brennecken,&nbsp;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}

{"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}

{"title":"Partition functions for non-commutative harmonic oscillators and related divergent series","authors":"Kazufumi Kimoto ,&nbsp;Masato Wakayama","doi":"10.1016/j.indag.2024.05.011","DOIUrl":"10.1016/j.indag.2024.05.011","url":null,"abstract":"<div><div><span><span>In the standard normalization, the eigenvalues of the quantum harmonic oscillator<span> are given by positive half-integers with the Hermite functions as eigenfunctions<span>. Thus, its spectral zeta function is essentially given by the Riemann zeta function. The heat kernel (or propagator) of the quantum harmonic oscillator (qHO) is given by the Mehler formula, and the </span></span></span>partition function is obtained by taking its trace. In general, the spectral zeta function of the given system is obtained by the Mellin transform of its partition function. In the case of non-commutative harmonic oscillators (NCHO), however, the heat kernel and partition functions are still unknown, although meromorphic continuation of the corresponding spectral zeta function and special values at positive integer points have been studied. On the other hand, explicit formulas for the heat kernel and partition function have been obtained for the quantum Rabi model (QRM), which is the simplest and most fundamental model for light and matter interaction in addition to having the NCHO as a covering model. In this paper, we propose a notion of the </span><em>quasi-partition function</em> for a quantum interaction model if the corresponding spectral zeta function can be meromorphically continued to the whole complex plane. The quasi-partition function for qHO and QRM actually gives the partition function. Assuming that this holds for the NCHO (currently a conjecture), we can find various interesting properties for the spectrum of the NCHO. Moreover, although we cannot expect any functional equation of the spectral zeta function for the quantum interaction models, we try to seek if there is some relation between the special values at positive and negative points. Attempting to seek this, we encounter certain divergent series expressing formally the Hurwitz zeta function by calculating integrals of the partition functions. We then give two interpretations of these divergent series by the Borel summation and <span><math><mi>p</mi></math></span>-adically convergent series defined by the <span><math><mi>p</mi></math></span>-adic Hurwitz zeta function.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 302-336"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550825","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}

{"title":"Holomorphic Laplacian on the Lie ball and the Penrose transform","authors":"Hideko Sekiguchi","doi":"10.1016/j.indag.2024.04.004","DOIUrl":"10.1016/j.indag.2024.04.004","url":null,"abstract":"<div><div>We prove that any holomorphic function <span><math><mi>f</mi></math></span> on the Lie ball of even dimension satisfying <span><math><mrow><mi>Δ</mi><mi>f</mi><mo>=</mo><mn>0</mn></mrow></math></span><span> is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from our setting that the line bundle parameter is </span><em>outside the good range</em>, we use some techniques from algebraic representation theory.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 114-123"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140928520","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}