Locally compact abelian groups with symplectic self-duality

Abelian subgroups and semisimple elements in $2$-modular representations of the symplectic group $Sp_{2n}(2)$

We determine the irreducible 2-modular representations of the symplectic group $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant information on the existence of eigenvalue 1 of elements of $G$ in a given irreducible representation of $G$.

阿贝尔群的度量对偶性

Metric duality for Abelian groups

该研究在拓扑阿贝尔群范畴中引入拓展经典赋范向量空间对偶的度量对偶概念，证明Polish LCA群有自反紧合度量，所有LCA群都有自反度量结构。

关于阿贝尔拓扑群 Pontryagin 对偶性的若干注记

Some notes on Pontryagin duality of abelian topological groups

本文探讨了阿贝尔pro-Lie群中Pontryagin对偶性相关的若干问题，分析了该领域内的核心方法与主要贡献。

Pontryagin duality for Abelian $s$- and $sb$-groups

The main goal of the article is to study the Pontryagin duality for Abelian $s$- and $sb$-groups. Let $G$ be an infinite Abelian group and $X$ be the dual group of the discrete group $G_d$. We show that a dense subgroup $H$ of $X$ is $\mathfrak{g}$-closed iff $H$ algebraically is the dual group of $G$ endowed with some maximally almost periodic $s$-topology. Every reflexive Polish Abelian group is $\mathfrak{g}$-closed in its Bohr compactification. If a $s$-topology $τ$ on a countably infinite Abelian group $G$ is generated by a countable set of convergent sequences, then the dual group of $(G…

Pontryagin duality between compact and discrete abelian inverse monoids

For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …

Abelian duality and propagation of resonance

We explore the relationship between a certain "abelian duality" property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the "EPY property." The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups,…