Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a16
Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He
$defDerL{operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $operatorname{Der}(L)$ on the $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $Gamma (wedge^bullet A^vee otimes L/A)$, and for this reason we elect to call the $DerL$-action internal symmetry of $Gamma (wedge^bullet A^vee otimes L/A)$.
{"title":"Internal symmetry of the $L_{leqslant 3}$ algebra arising from a Lie pair","authors":"Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He","doi":"10.4310/pamq.2023.v19.n4.a16","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a16","url":null,"abstract":"$defDerL{operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $operatorname{Der}(L)$ on the $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $Gamma (wedge^bullet A^vee otimes L/A)$, and for this reason we elect to call the $DerL$-action internal symmetry of $Gamma (wedge^bullet A^vee otimes L/A)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"263 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517072","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 : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a8
Rebecca Goldin, Julianna Tymoczko
Hessenberg varieties $mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the $T$-stable Hessenberg varieties, we identify several that are GKM spaces, meaning $T$ acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type $A_{n-1}$ and in general Lie type.
{"title":"Which Hessenberg varieties are GKM?","authors":"Rebecca Goldin, Julianna Tymoczko","doi":"10.4310/pamq.2023.v19.n4.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a8","url":null,"abstract":"Hessenberg varieties $mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the $T$-stable Hessenberg varieties, we identify several that are <i>GKM spaces</i>, meaning $T$ acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type $A_{n-1}$ and in general Lie type.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"195 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495291","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 : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a3
Ana Cannas da Silva
This paper, written for differential geometers, shows how to deduce the reciprocity laws of Dedekind and Rademacher, as well as $n$-dimensional generalizations of these, from the Atiyah–Bott–Lefschetz formula, by applying this formula to appropriate elliptic complexes on weighted projective spaces.
{"title":"Dedekind sums via Atiyah–Bott–Lefschetz","authors":"Ana Cannas da Silva","doi":"10.4310/pamq.2023.v19.n4.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a3","url":null,"abstract":"This paper, written for differential geometers, shows how to deduce the reciprocity laws of Dedekind and Rademacher, as well as $n$-dimensional generalizations of these, from the Atiyah–Bott–Lefschetz formula, by applying this formula to appropriate elliptic complexes on weighted projective spaces.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"194 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495296","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 : 2023-04-07DOI: 10.4310/pamq.2023.v19.n2.a1
Oliver Braunling
Usually the boundary map in $K$-theory localization only gives the tame symbol at $K_2$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at $p=2$. Our idea is to pinch singularities near the ramification locus. This fattens up $K$-theory and makes the wild symbol visible as a boundary map.
{"title":"Hilbert reciprocity using $K$-theory localization","authors":"Oliver Braunling","doi":"10.4310/pamq.2023.v19.n2.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a1","url":null,"abstract":"Usually the boundary map in $K$-theory localization only gives the tame symbol at $K_2$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at $p=2$. Our idea is to pinch singularities near the ramification locus. This fattens up $K$-theory and makes the wild symbol visible as a boundary map.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"198 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495284","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 : 2023-01-01DOI: 10.4310/pamq.2023.v19.n1.a8
Jens Funke, S. Kudla
{"title":"Indefinite theta series: the case of an $N$-gon","authors":"Jens Funke, S. Kudla","doi":"10.4310/pamq.2023.v19.n1.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a8","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526678","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 : 2023-01-01DOI: 10.4310/pamq.2023.v19.n2.a7
Lu Zhao
{"title":"Sum expressions for $p$-adic Hecke $L$-functions of totally real fields","authors":"Lu Zhao","doi":"10.4310/pamq.2023.v19.n2.a7","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a7","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526796","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 : 2023-01-01DOI: 10.4310/pamq.2023.v19.n2.a9
Weidong Wang, Yanping Zhou
{"title":"Some inequalities for the dual $p$-quermassintegrals","authors":"Weidong Wang, Yanping Zhou","doi":"10.4310/pamq.2023.v19.n2.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a9","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526842","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}
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