A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(Gtimes H)/{rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${rm dim,Hom}_H(pi|_H,tau)
{"title":"Symmetry Breaking Operators for Strongly Spherical Reductive Pairs","authors":"Jan Frahm","doi":"10.4171/prims/59-2-2","DOIUrl":"https://doi.org/10.4171/prims/59-2-2","url":null,"abstract":"A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(Gtimes H)/{rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${rm dim,Hom}_H(pi|_H,tau)<infty$ for all smooth admissible representations $pi$ of $G$ and $tau$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${rm Hom}_H(pi|_H,tau)$ for $pi$ and $tau$ spherical principal series representations of $G$ and $H$. These so-called symmetry breaking operators depend holomorphically on the induction parameters and we further show that they generically span the space ${rm Hom}_H(pi|_H,tau)$. In the special case of multiplicity one pairs we extend our construction to vector-valued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove an early version of the Gross-Prasad conjecture for complex orthogonal groups, and also provide lower bounds for the dimension of the space of Shintani functions.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical Néron–Severi lattice of $Y$ induced by $f$. We combine a modification of McMullen’s method with Borcherds’ method to prove that the modulo-$2$ reduction $(p_f(x) bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $Phi_m$, where $m leq 9$ and $m$ is odd. We study Enriques surfaces that realizevmodulo-$2$ reductions of $Phi_7$, $Phi_9$ and show that each of the five polynomials $(Phi_m(x) bmod 2)$ is a factor of the modulo-$2$ reduction $(p_f(x) bmod 2)$ for a complex Enriques surface.
{"title":"On Characteristic Polynomials of Automorphisms of Enriques Surfaces","authors":"Simon Brandhorst, Sławomir Rams, Ichiro Shimada","doi":"10.4171/prims/59-3-7","DOIUrl":"https://doi.org/10.4171/prims/59-3-7","url":null,"abstract":"Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical Néron–Severi lattice of $Y$ induced by $f$. We combine a modification of McMullen’s method with Borcherds’ method to prove that the modulo-$2$ reduction $(p_f(x) bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $Phi_m$, where $m leq 9$ and $m$ is odd. We study Enriques surfaces that realizevmodulo-$2$ reductions of $Phi_7$, $Phi_9$ and show that each of the five polynomials $(Phi_m(x) bmod 2)$ is a factor of the modulo-$2$ reduction $(p_f(x) bmod 2)$ for a complex Enriques surface.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $sigma$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton–Jacobi–Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow us to define the generators in a weak sense.
{"title":"Convex Monotone Semigroups on Lattices of Continuous Functions","authors":"Robert Denk, Michael Kupper, Max Nendel","doi":"10.4171/prims/59-2-4","DOIUrl":"https://doi.org/10.4171/prims/59-2-4","url":null,"abstract":"We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $sigma$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton–Jacobi–Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow us to define the generators in a weak sense.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to Hörmander’s support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.
{"title":"Quasianalytic Functionals and Ultradistributions as Boundary Values of Harmonic Functions","authors":"Andreas Debrouwere, Jasson Vindas","doi":"10.4171/prims/59-3-8","DOIUrl":"https://doi.org/10.4171/prims/59-3-8","url":null,"abstract":"We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to Hörmander’s support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p$ be a prime number. In the present paper, we consider a certain pro-$p$ analogue of the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over $p$-adic local fields [i.e., finite extensions of the field of $p$-adic numbers $mathbb{Q}_p$] obtained by Mochizuki. This research was motivated by Higashiyama’s recent work on the pro-$p$ analogue of the semi-absolute version of the Grothendieck conjecture for configuration spaces [of dimension $geq 2$] associated to hyperbolic curves over generalized sub-$p$-adic fields [i.e., subfields of finitely generated extensions of the completion of the maximal unramified extension of $mathbb{Q}_p$].
设p是质数。在本文中，我们考虑了$p$-一元局部域上光滑变异的基本群之间同构的半绝对性的一类亲$p$类比。， Mochizuki得到的$p$-进数$mathbb{Q}_p$]域的有限扩展。这项研究的动机是源于Higashiyama最近的工作，该工作是关于与广义sub- p -adic域上的双曲曲线相关的位形空间的格罗腾迪克猜想的半绝对版本的亲p模拟。， $mathbb{Q}_p$]的最大无分支扩展补全的有限生成扩展的子域。
{"title":"On the Semi-absoluteness of Isomorphisms between the Pro-$p$ Arithmetic Fundamental Groups of Smooth Varieties","authors":"Shota Tsujimura","doi":"10.4171/prims/59-3-3","DOIUrl":"https://doi.org/10.4171/prims/59-3-3","url":null,"abstract":"Let $p$ be a prime number. In the present paper, we consider a certain pro-$p$ analogue of the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over $p$-adic local fields [i.e., finite extensions of the field of $p$-adic numbers $mathbb{Q}_p$] obtained by Mochizuki. This research was motivated by Higashiyama’s recent work on the pro-$p$ analogue of the semi-absolute version of the Grothendieck conjecture for configuration spaces [of dimension $geq 2$] associated to hyperbolic curves over generalized sub-$p$-adic fields [i.e., subfields of finitely generated extensions of the completion of the maximal unramified extension of $mathbb{Q}_p$].","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.
{"title":"Fractional Kolmogorov Operator and Desingularizing Weights","authors":"Damir Kinzebulatov, Yuliy A. Semënov","doi":"10.4171/prims/59-2-3","DOIUrl":"https://doi.org/10.4171/prims/59-2-3","url":null,"abstract":"We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we define the affine super Yangian $Y_{varepsilon_1,varepsilon_2}(widehat{mathfrak{sl}}(m|n))$ with a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from $Y_{varepsilon_1,varepsilon_2}(widehat{mathfrak{sl}}(m|n))$ to the completion of the universal enveloping algebra of $widehat{mathfrak{gl}}(m|n)$.
{"title":"Construction of the Affine Super Yangian","authors":"Mamoru Ueda","doi":"10.4171/prims/59-3-1","DOIUrl":"https://doi.org/10.4171/prims/59-3-1","url":null,"abstract":"In this paper, we define the affine super Yangian $Y_{varepsilon_1,varepsilon_2}(widehat{mathfrak{sl}}(m|n))$ with a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from $Y_{varepsilon_1,varepsilon_2}(widehat{mathfrak{sl}}(m|n))$ to the completion of the universal enveloping algebra of $widehat{mathfrak{gl}}(m|n)$.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136292361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The deduction starts with the (non-relativistic) one-photon Hilbert space $mathcal{H}$, equipped with the one-photon Hamiltonian and basic symmetry generators, as the only input information. We recall in the functorially associated boson Fock space the multi-photon dynamics and symmetry transformations, as well as the field operator (as the scaled self-adjoint part of the creation operator) and the q(uasi)-classical states. There is no reference to a presupposed classical Maxwell theory. By abstraction, we go over to the algebraic formulation of the multi-photon theory in terms of a C*-Weyl algebra. Its test function space $Esubset mathcal{H}$ is constructed as a nuclear Fréchet space, in which – via infrared damping – the dynamics and symmetries are nuclear continuous and their generators bounded. Each w*-closed, singular subspace of the continuous dual $E'$ determines non-Fock coherent states and their mixtures lead to a representation von Neumann algebra with non-trivial center. The symmetry generators restricted to the center can be transformed into the Maxwell form by means of a symplectic transformation and involve the well-known conservation quantities of electrodynamics. This identifies the central part of the represented photon field operator as composed of the two classical canonical electrodynamic field components. We have obtained, therefore, in free space a kind of fusion of the multi-photon theory and the Maxwell theory of transverse electrodynamic fields, where the latter arise as derived quantities. By means of a Bogoliubov transformation one also gets a fusion of the quantized with the classical Maxwell theory, deduced from the photon concept. A sketch of non-relativistic gauging is added in the appendix to gain longitudinal, cohomological, and scalar potentials.
{"title":"Classical and Quantized Maxwell Fields Deduced from Algebraic Many-Photon Theory","authors":"Alfred Rieckers","doi":"10.4171/prims/59-2-1","DOIUrl":"https://doi.org/10.4171/prims/59-2-1","url":null,"abstract":"The deduction starts with the (non-relativistic) one-photon Hilbert space $mathcal{H}$, equipped with the one-photon Hamiltonian and basic symmetry generators, as the only input information. We recall in the functorially associated boson Fock space the multi-photon dynamics and symmetry transformations, as well as the field operator (as the scaled self-adjoint part of the creation operator) and the q(uasi)-classical states. There is no reference to a presupposed classical Maxwell theory. By abstraction, we go over to the algebraic formulation of the multi-photon theory in terms of a C*-Weyl algebra. Its test function space $Esubset mathcal{H}$ is constructed as a nuclear Fréchet space, in which – via infrared damping – the dynamics and symmetries are nuclear continuous and their generators bounded. Each w*-closed, singular subspace of the continuous dual $E'$ determines non-Fock coherent states and their mixtures lead to a representation von Neumann algebra with non-trivial center. The symmetry generators restricted to the center can be transformed into the Maxwell form by means of a symplectic transformation and involve the well-known conservation quantities of electrodynamics. This identifies the central part of the represented photon field operator as composed of the two classical canonical electrodynamic field components. We have obtained, therefore, in free space a kind of fusion of the multi-photon theory and the Maxwell theory of transverse electrodynamic fields, where the latter arise as derived quantities. By means of a Bogoliubov transformation one also gets a fusion of the quantized with the classical Maxwell theory, deduced from the photon concept. A sketch of non-relativistic gauging is added in the appendix to gain longitudinal, cohomological, and scalar potentials.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extended Affine Root Supersystems of Types $C(I, J)$ and $BC(1, 1)$","authors":"M. Yousofzadeh","doi":"10.4171/prims/59-1-3","DOIUrl":"https://doi.org/10.4171/prims/59-1-3","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44022980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integrality of boldmath$v$-adic Multiple Zeta Values","authors":"Yen-Tsung Chen","doi":"10.4171/prims/59-1-4","DOIUrl":"https://doi.org/10.4171/prims/59-1-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43286720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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