Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.05.002
Mengyuan Cao, Monica Nevins, Hadi Salmasian
Let be either the Lie superalgebra where or the Lie superalgebra where . Furthermore, let be the -module defined by in the former case and in the latter case. Associated to there exists a distinguished basis of Capelli operators , naturally indexed by a set of hook partitions , for the subalgebra of -invariants in the superalgebra of superdifferential operators on .
Let be a Borel subalgebra of . We compute eigenvalues of the on the irreducible -submodules of and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev–Veselov at suitable affine functions of the -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.
{"title":"The refined solution to the Capelli eigenvalue problem for gl(m|n)⊕gl(m|n) and gl(m|2n)","authors":"Mengyuan Cao, Monica Nevins, 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}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.05.012
Jan Frahm , Gestur Ólafsson , Bent Ørsted
For every simple Hermitian Lie group , we consider a certain maximal parabolic subgroup whose unipotent radical is either abelian (if is of tube type) or two-step nilpotent (if is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of , the space of square-integrable sections of the homogeneous vector bundle over associated with an irreducible unitary representation of . Assuming that the central character of is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of into and show that the multiplicities are equal to the dimensions of the lowest -types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of . This kernel function carries all information about the holomorphic discrete series embedding, the lowest -type as functions on , as well as the associated Whittaker vectors.
{"title":"The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups","authors":"Jan Frahm , Gestur Ólafsson , 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}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.04.005
Maarten Solleveld
Let be a reductive group over a non-archimedean local field . Consider an arbitrary Bernstein block in the category of complex smooth -representations. In earlier work the author showed that there exists an affine Hecke algebra whose category of right modules is closely related to . In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.
In this paper we study the -parameters of the affine Hecke algebras . We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.
Lusztig conjectured that the -parameters are always integral powers of the cardinality of the residue field of , and that they coincide with the -parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple -adic groups, and we prove it for most of those.
{"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}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.05.001
K.-U. Bux , J. Hilgert , T. Weich
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.
{"title":"Spectral correspondences for finite graphs without dead ends","authors":"K.-U. Bux , J. Hilgert , 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}
Pub Date : 2025-01-01DOI: 10.1016/j.indag.2024.04.001
Aloysius G. Helminck , Gerardus F. Helminck
<div><div>Inside the algebra <span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi></mrow></math></span>-matrices with coefficients from a commutative <span><math><mi>ℂ</mi></math></span>-algebra <span><math><mi>R</mi></math></span> that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> 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 <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> have to satisfy are determined by the decomposition of <span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span><span>-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 </span><span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span><span>-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the </span><span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy are constructed. There is a real analogue of the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy called the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy. It consists of a deformation of a commutative Lie algebra <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>s</mi></mro
{"title":"A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian Z×Z-matrices","authors":"Aloysius G. Helminck , Gerardus F. Helminck","doi":"10.1016/j.indag.2024.04.001","DOIUrl":"10.1016/j.indag.2024.04.001","url":null,"abstract":"<div><div>Inside the algebra <span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi></mrow></math></span>-matrices with coefficients from a commutative <span><math><mi>ℂ</mi></math></span>-algebra <span><math><mi>R</mi></math></span> that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> 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 <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> have to satisfy are determined by the decomposition of <span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span><span>-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 </span><span><math><mrow><mi>L</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>Z</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span><span>-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the </span><span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy are constructed. There is a real analogue of the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy called the <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>-hierarchy. It consists of a deformation of a commutative Lie algebra <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>s</mi></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}
Pub Date : 2024-11-01DOI: 10.1016/j.indag.2024.06.001
In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple -values (AMMVs), forming a -subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman’s multiple -values, Kaneko–Tsumura’s multiple -values, and the multiple -values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by two of us. We also investigate several alternating multiple - and -values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.
{"title":"Alternating multiple mixed values: Regularization, special values, parity, and dimension conjectures","authors":"","doi":"10.1016/j.indag.2024.06.001","DOIUrl":"10.1016/j.indag.2024.06.001","url":null,"abstract":"<div><div>In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple <span><math><mi>M</mi></math></span>-values (AMMVs), forming a <span><math><mrow><mi>Q</mi><mrow><mo>[</mo><mi>i</mi><mo>]</mo></mrow></mrow></math></span>-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman’s multiple <span><math><mi>t</mi></math></span>-values, Kaneko–Tsumura’s multiple <span><math><mi>T</mi></math></span>-values, and the multiple <span><math><mi>S</mi></math></span>-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by two of us. We also investigate several alternating multiple <span><math><mi>S</mi></math></span>- and <span><math><mi>T</mi></math></span>-values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1212-1248"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504919","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 : 2024-11-01DOI: 10.1016/j.indag.2024.06.003
Our aim is to find all the prime numbers such that has at most two different prime factors, for all the odd integers such that . We solve entirely the cases , using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers such that has at most two different prime factors, for all the odd integers such that .
{"title":"Class numbers, Ono invariants and some interesting primes","authors":"","doi":"10.1016/j.indag.2024.06.003","DOIUrl":"10.1016/j.indag.2024.06.003","url":null,"abstract":"<div><div>Our aim is to find all the prime numbers <span><math><mi>p</mi></math></span> such that <span><math><mrow><mi>p</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> has at most two different prime factors, for all the odd integers <span><math><mi>x</mi></math></span> such that <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>p</mi></mrow></math></span>. We solve entirely the cases <span><math><mrow><mi>p</mi><mo>≡</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>8</mn><mo>)</mo></mrow></mrow></math></span>, using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case <span><math><mrow><mi>p</mi><mo>≡</mo><mn>7</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>8</mn><mo>)</mo></mrow></mrow></math></span> is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value <span><math><mrow><mi>p</mi><mo>≡</mo><mn>7</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>8</mn><mo>)</mo></mrow></mrow></math></span> besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers <span><math><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></math></span> such that <span><math><mrow><mi>n</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> has at most two different prime factors, for all the odd integers <span><math><mi>x</mi></math></span> such that <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>n</mi></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1249-1258"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504918","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 : 2024-11-01DOI: 10.1016/j.indag.2024.05.010
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
{"title":"Integral expressions for Schur multiple zeta values","authors":"","doi":"10.1016/j.indag.2024.05.010","DOIUrl":"10.1016/j.indag.2024.05.010","url":null,"abstract":"<div><div>Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1197-1211"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528966","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 : 2024-11-01DOI: 10.1016/j.indag.2024.05.003
We construct large families of simple modules for untwisted affine Lie algebras using induction from one-dimensional modules over nilpotent loop subalgebras. We also show that the vector space of the first self-extensions for these module has uncountable dimension and that generic tensor products of these modules are simple.
{"title":"Simple modules for untwisted affine Lie algebras induced from nilpotent loop subalgebras","authors":"","doi":"10.1016/j.indag.2024.05.003","DOIUrl":"10.1016/j.indag.2024.05.003","url":null,"abstract":"<div><div>We construct large families of simple modules for untwisted affine Lie algebras using induction from one-dimensional modules over nilpotent loop subalgebras. We also show that the vector space of the first self-extensions for these module has uncountable dimension and that generic tensor products of these modules are simple.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1138-1148"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060172","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 : 2024-11-01DOI: 10.1016/j.indag.2024.05.006
In this document we consider an exact sequence of group varieties over an algebraically closed field. We show that for a prime there exists an isomorphism of graded -algebras that is compatible with pullback homomorphisms of endomorphisms that stabilize .
{"title":"Functorial splitting of l-adic cohomology of an extension of group varieties","authors":"","doi":"10.1016/j.indag.2024.05.006","DOIUrl":"10.1016/j.indag.2024.05.006","url":null,"abstract":"<div><div>In this document we consider an exact sequence of group varieties <span><math><mrow><mi>e</mi><mo>→</mo><mi>N</mi><mo>→</mo><mi>G</mi><mo>→</mo><mi>Q</mi><mo>→</mo></mrow></math></span> <span><math><mi>e</mi></math></span> over an algebraically closed field. We show that for <span><math><mrow><mi>l</mi><mo>≠</mo><mi>char</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> a prime there exists an isomorphism of graded <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>-algebras <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>́</mo></mrow></mover><mi>t</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></mrow><mo>≅</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>́</mo></mrow></mover><mi>t</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mrow><mo>(</mo><mi>N</mi><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></mrow><msub><mrow><mo>⊗</mo></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><msubsup><mrow><mi>H</mi></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>́</mo></mrow></mover><mi>t</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> that is compatible with pullback homomorphisms <span><math><msup><mrow><mi>φ</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> of endomorphisms <span><math><mrow><mi>φ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow></math></span> that stabilize <span><math><mi>N</mi></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1185-1196"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504921","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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