Indagationes Mathematicae-New Series最新文献

{"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}

{"title":"The refined solution to the Capelli eigenvalue problem for gl(m|n)⊕gl(m|n) and gl(m|2n)","authors":"Mengyuan Cao,&nbsp;Monica Nevins,&nbsp;Hadi Salmasian","doi":"10.1016/j.indag.2024.05.002","DOIUrl":"10.1016/j.indag.2024.05.002","url":null,"abstract":"<div><div>Let <span><math><mi>g</mi></math></span> be either the Lie superalgebra <span><math><mrow><mi>gl</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>⊕</mo><mi>gl</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>V</mi><mo>≔</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>m</mi><mo>|</mo><mi>n</mi></mrow></msup></mrow></math></span> or the Lie superalgebra <span><math><mrow><mi>gl</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>V</mi><mo>≔</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>m</mi><mo>|</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></math></span>. Furthermore, let <span><math><mi>W</mi></math></span> be the <span><math><mi>g</mi></math></span>-module defined by <span><math><mrow><mi>W</mi><mo>≔</mo><mi>V</mi><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> in the former case and <span><math><mrow><mi>W</mi><mo>≔</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></math></span> in the latter case. Associated to <span><math><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>W</mi><mo>)</mo></mrow></math></span> there exists a distinguished basis of <em>Capelli operators</em> <span><math><msub><mrow><mrow><mo>{</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo>}</mo></mrow></mrow><mrow><mi>λ</mi><mo>∈</mo><mi>Ω</mi></mrow></msub></math></span>, naturally indexed by a set of hook partitions <span><math><mi>Ω</mi></math></span>, for the subalgebra of <span><math><mi>g</mi></math></span>-invariants in the superalgebra <span><math><mrow><mi>PD</mi><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow></mrow></math></span> of superdifferential operators on <span><math><mi>W</mi></math></span>.</div><div>Let <span><math><mi>b</mi></math></span> be a Borel subalgebra of <span><math><mi>g</mi></math></span>. We compute eigenvalues of the <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow></msup></math></span> on the irreducible <span><math><mi>g</mi></math></span>-submodules of <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow></mrow></math></span> and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev–Veselov at suitable affine functions of the <span><math><mi>b</mi></math></span>-highest weight. While the former case is straightforward, the latter is significantly more complex. This generalizes a result by Sahi, Salmasian and Serganova for these cases, where such formulas were given for a fixed choice of Borel subalgebra.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 218-244"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141034907","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":"The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups","authors":"Jan Frahm ,&nbsp;Gestur Ólafsson ,&nbsp;Bent Ørsted","doi":"10.1016/j.indag.2024.05.012","DOIUrl":"10.1016/j.indag.2024.05.012","url":null,"abstract":"<div><div>For every simple Hermitian Lie group <span><math><mi>G</mi></math></span>, we consider a certain maximal parabolic subgroup whose unipotent radical <span><math><mi>N</mi></math></span> is either abelian (if <span><math><mi>G</mi></math></span> is of tube type) or two-step nilpotent (if <span><math><mi>G</mi></math></span> is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>/</mo><mi>N</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span>, the space of square-integrable sections of the homogeneous vector bundle over <span><math><mrow><mi>G</mi><mo>/</mo><mi>N</mi></mrow></math></span> associated with an irreducible unitary representation <span><math><mi>ω</mi></math></span> of <span><math><mi>N</mi></math></span>. Assuming that the central character of <span><math><mi>ω</mi></math></span> is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of <span><math><mi>G</mi></math></span> into <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>/</mo><mi>N</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span> and show that the multiplicities are equal to the dimensions of the lowest <span><math><mi>K</mi></math></span>-types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of <span><math><mi>G</mi></math></span>. This kernel function carries all information about the holomorphic discrete series embedding, the lowest <span><math><mi>K</mi></math></span>-type as functions on <span><math><mrow><mi>G</mi><mo>/</mo><mi>N</mi></mrow></math></span>, as well as the associated Whittaker vectors.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 337-356"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504920","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":"Parameters of Hecke algebras for Bernstein components of p-adic groups","authors":"Maarten Solleveld","doi":"10.1016/j.indag.2024.04.005","DOIUrl":"10.1016/j.indag.2024.04.005","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a reductive group over a non-archimedean local field <span><math><mi>F</mi></math></span>. Consider an arbitrary Bernstein block <span><math><mrow><mi>Rep</mi><msup><mrow><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup></mrow></math></span> in the category of complex smooth <span><math><mi>G</mi></math></span>-representations. In earlier work the author showed that there exists an affine Hecke algebra <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>O</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> whose category of right modules is closely related to <span><math><mrow><mi>Rep</mi><msup><mrow><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup></mrow></math></span>. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.</div><div>In this paper we study the <span><math><mi>q</mi></math></span>-parameters of the affine Hecke algebras <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>O</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.</div><div>Lusztig conjectured that the <span><math><mi>q</mi></math></span>-parameters are always integral powers of the cardinality of the residue field of <span><math><mi>F</mi></math></span>, and that they coincide with the <span><math><mi>q</mi></math></span>-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple <span><math><mi>p</mi></math></span>-adic groups, and we prove it for most of those.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 124-170"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140778482","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":"Spectral correspondences for finite graphs without dead ends","authors":"K.-U. Bux ,&nbsp;J. Hilgert ,&nbsp;T. Weich","doi":"10.1016/j.indag.2024.05.001","DOIUrl":"10.1016/j.indag.2024.05.001","url":null,"abstract":"<div><div>We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non-backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 188-217"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513687","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":"A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian Z×Z-matrices","authors":"Aloysius G. Helminck ,&nbsp;Gerardus F. Helminck","doi":"10.1016/j.indag.2024.04.001","DOIUrl":"10.1016/j.indag.2024.04.001","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Inside the algebra &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-matrices with coefficients from a commutative &lt;span&gt;&lt;math&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of finite band skew hermitian matrices that is different from the Lie subalgebras that were deformed at the discrete KP hierarchy and its strict version. The evolution equations that the deformed generators of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; have to satisfy are determined by the decomposition of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;-hierarchy. We show that the projections of a solution satisfy zero curvature relations and that it suffices to solve an associated Cauchy problem. Solutions of this type can be obtained by finding appropriate vectors in the &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-hierarchy are constructed. There is a real analogue of the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-hierarchy called the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-hierarchy. It consists of a deformation of a commutative Lie algebra &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mro","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 1","pages":"Pages 42-60"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609803","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 an estimate on Götzky’s domain","authors":"Dávid Tóth","doi":"10.1016/j.indag.2024.12.004","DOIUrl":"10.1016/j.indag.2024.12.004","url":null,"abstract":"<div><div>A fundamental domain <span><math><mrow><mi>F</mi><mo>⊂</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> for the Hilbert modular group belonging to the quadratic number field <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> was constructed by Götzky almost a hundred years ago. He also gave a lower bound for the height <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> of the points <span><math><mrow><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>i</mi><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>i</mi><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>∈</mo><mi>F</mi></mrow></math></span>. Later Gundlach used analogous domains and estimates for other fields as well to give a complete list of totally elliptic conjugacy classes in some Hilbert modular groups, while not long ago Deutsch analyzed two of these domains by numerical computations and stated some conjectures about them. We prove one of these by giving a sharp lower bound for the height of the points of Götzky’s domain.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 4","pages":"Pages 1096-1111"},"PeriodicalIF":0.5,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270398","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 evaluations of multiple S- and T-values of the form S(2(−),1,…,1,1(−)) and T(2(−),1,…,1,1(−))","authors":"Steven Charlton","doi":"10.1016/j.indag.2024.12.001","DOIUrl":"10.1016/j.indag.2024.12.001","url":null,"abstract":"<div><div>Xu, Yan and Zhao showed that in even weight, the multiple <span><math><mi>T</mi></math></span>-value <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mover><mrow><mn>1</mn></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> is a polynomial in <span><math><mrow><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo><mi>π</mi></mrow></math></span>, Riemann zeta-values, and Dirichlet beta-values. Based on low-weight examples, they conjectured that <span><math><mrow><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> does not appear in the evaluation. We show that their conjecture is correct, and in fact follows largely from various earlier results of theirs. More precisely, we derive explicit formulae for <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mover><mrow><mn>1</mn></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> in even weight and <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mover><mrow><mn>1</mn></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> in odd weight via generating series calculations. We also resolve another conjecture of theirs on the evaluations of <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mover><mrow><mn>2</mn></mrow><mo>¯</mo></mover><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mover><mrow><mn>1</mn></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mover><mrow><mn>2</mn></mrow><mo>¯</mo></mover><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mover><mrow><mn>2</mn></mrow><mo>¯</mo></mover><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mover><mrow><mn>1</mn></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> in even weight, by way of calculations involving Goncharov’s theory of iterated integrals and multiple polylogarithms.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 4","pages":"Pages 1055-1083"},"PeriodicalIF":0.5,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270396","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":"The Waldschmidt constant of special fat flat subschemes in PN","authors":"Hassan Haghighi,&nbsp;Mohammad Mosakhani","doi":"10.1016/j.indag.2024.12.003","DOIUrl":"10.1016/j.indag.2024.12.003","url":null,"abstract":"<div><div>The purpose of this paper is to construct some special kind of subschemes in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, which we call them “fat flat subschemes” and compute their Waldschmidt constants. These subschemes are constructed by adding, in a particular way, a finite number of linear subspaces of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> of many different dimensions to a star configuration in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, with arbitrary preassigned multiplicities to each one of these linear subspaces, as well as the star configuration. Among other things, it will be shown that for every positive integer <span><math><mi>d</mi></math></span>, there are infinitely many fat flat subschemes in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with the Waldschmidt constant equal to <span><math><mi>d</mi></math></span>. In addition to this, for any two integers <span><math><mrow><mn>1</mn><mo>≤</mo><mi>a</mi><mo>&lt;</mo><mi>b</mi></mrow></math></span>, we also construct a fat flat subscheme of the above type in some projective space <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>M</mi></mrow></msup></math></span>, whose Waldschmidt constant is equal to <span><math><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow></math></span>. In addition to these, all non-reduced fat points subschemes <span><math><mi>Z</mi></math></span> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with the Waldschmidt constants less than <span><math><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></math></span> are classified.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 4","pages":"Pages 1084-1095"},"PeriodicalIF":0.5,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270397","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":"The chromatic number of 4-dimensional lattices","authors":"Frank Vallentin,&nbsp;Stephen Weißbach,&nbsp;Marc Christian Zimmermann","doi":"10.1016/j.indag.2024.11.006","DOIUrl":"10.1016/j.indag.2024.11.006","url":null,"abstract":"<div><div>The chromatic number of a lattice in <span><math><mi>n</mi></math></span>-dimensional Euclidean space is defined as the chromatic number of its Voronoi graph. The Voronoi graph is the Cayley graph on the lattice having the strict Voronoi vectors as generators. In this paper we determine the chromatic number of all 4-dimensional lattices.</div><div>To achieve this we use the known classification of 52 parallelohedra in dimension 4. These 52 geometric types yield 16 combinatorial types of relevant Voronoi graphs. We discuss a systematic approach to checking for isomorphism of Cayley graphs of lattices.</div><div>Lower bounds for the chromatic number are obtained from choosing appropriate small finite induced subgraphs of the Voronoi graphs. Matching upper bounds are derived from periodic colorings. To determine the chromatic numbers of these finite graphs, we employ a SAT solver.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 4","pages":"Pages 988-1004"},"PeriodicalIF":0.5,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270458","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}