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}
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