Pub Date : 2020-03-13DOI: 10.2140/tunis.2021.3.551
J. Clerc
A new formula is obtained for the holomorphic bi-differential operators on tube-type domains which are associated to the decomposition of the tensor product of two scalar holomorphic representations, thus generalizing the classical Rankin-Cohen brackets. The formula involves a family of polynomials of several variables which may be considered as a (weak) generalization of the classical Jacobi polynomials.
{"title":"Rankin-Cohen brackets on tube-type domains","authors":"J. Clerc","doi":"10.2140/tunis.2021.3.551","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.551","url":null,"abstract":"A new formula is obtained for the holomorphic bi-differential operators on tube-type domains which are associated to the decomposition of the tensor product of two scalar holomorphic representations, thus generalizing the classical Rankin-Cohen brackets. The formula involves a family of polynomials of several variables which may be considered as a (weak) generalization of the classical Jacobi polynomials.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123073843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This map is independent of the choice of P. We show that for large q this map coincides with the known cohomological induction (whose definition involves a choice of P).
{"title":"On induction of class functions","authors":"G. Lusztig","doi":"10.1090/ERT/561","DOIUrl":"https://doi.org/10.1090/ERT/561","url":null,"abstract":"Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This map is independent of the choice of P. We show that for large q this map coincides with the known cohomological induction (whose definition involves a choice of P).","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123929797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $mathfrak{g}J(k)$ and $mathfrak{gl}_infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $mathfrak{g}J(k)$ and of the orthogonal and symplectic subalgebras.
{"title":"On Lie algebras of generalized Jacobi matrices","authors":"A. Fialowski, K. Iohara","doi":"10.4064/BC123-7","DOIUrl":"https://doi.org/10.4064/BC123-7","url":null,"abstract":"In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $mathfrak{g}J(k)$ and $mathfrak{gl}_infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $mathfrak{g}J(k)$ and of the orthogonal and symplectic subalgebras.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127441409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the following statement: Let $X=text{SL}_n(mathbb{Z})backslash text{SL}_n(mathbb{R})$, and consider the standard action of the diagonal group $A 0$ is some positive constant. Then any regular element $ain A$ acts on $mu$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $mathbb{Q}$, and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss.
{"title":"Positive Entropy Using Hecke Operators at a Single Place","authors":"Zvi Shem-Tov","doi":"10.1093/IMRN/RNAA235","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA235","url":null,"abstract":"We prove the following statement: Let $X=text{SL}_n(mathbb{Z})backslash text{SL}_n(mathbb{R})$, and consider the standard action of the diagonal group $A 0$ is some positive constant. Then any regular element $ain A$ acts on $mu$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $mathbb{Q}$, and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114633675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of Gelfand pair (G, K) can be generalized if we consider homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition of being a commutative homogeneous vector bundle. For the case in which G/K is a nilmanifold having square-integrable representations, in a previous article we determined a big family of commutative homogeneous vector bundles. In this paper, we complete that classification.
{"title":"On commutative homogeneous vector bundles attached to nilmanifolds","authors":"Roc'io D'iaz Mart'in, L. Saal","doi":"10.33044/REVUMA.1738","DOIUrl":"https://doi.org/10.33044/REVUMA.1738","url":null,"abstract":"The notion of Gelfand pair (G, K) can be generalized if we consider homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition of being a commutative homogeneous vector bundle. For the case in which G/K is a nilmanifold having square-integrable representations, in a previous article we determined a big family of commutative homogeneous vector bundles. In this paper, we complete that classification.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"429 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126082322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent results on the Racah algebra $mathcal{R}_n$ of rank $n - 2$ are reviewed. $mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $mathfrak{su}(1,1)$ in $mathcal{U}(mathfrak{su}(1,1))^{otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $ (n-1)$ - sphere and a number of interesting realizations are provided.
{"title":"The Racah algebra: An overview and recent\u0000 results","authors":"H. Bie, P. Iliev, W. Vijver, L. Vinet","doi":"10.1090/conm/768/15450","DOIUrl":"https://doi.org/10.1090/conm/768/15450","url":null,"abstract":"Recent results on the Racah algebra $mathcal{R}_n$ of rank $n - 2$ are reviewed. $mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $mathfrak{su}(1,1)$ in $mathcal{U}(mathfrak{su}(1,1))^{otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $ (n-1)$ - sphere and a number of interesting realizations are provided.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124908696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-27DOI: 10.1142/s021949882250147x
R. Jawad, N. Snashall
Given a finite-dimensional algebra $Lambda$ and $A geqslant 1$, we construct a new algebra $tilde{Lambda}_A$, called the stretched algebra, and relate the homological properties of $Lambda$ and $tilde{Lambda}_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $Lambda$ has (Fg) if and only if $tilde{Lambda}_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $Lambda$ is a $d$-Koszul algebra for some $d geqslant 2$.
{"title":"Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras","authors":"R. Jawad, N. Snashall","doi":"10.1142/s021949882250147x","DOIUrl":"https://doi.org/10.1142/s021949882250147x","url":null,"abstract":"Given a finite-dimensional algebra $Lambda$ and $A geqslant 1$, we construct a new algebra $tilde{Lambda}_A$, called the stretched algebra, and relate the homological properties of $Lambda$ and $tilde{Lambda}_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $Lambda$ has (Fg) if and only if $tilde{Lambda}_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $Lambda$ is a $d$-Koszul algebra for some $d geqslant 2$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130941966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smalo(HSR) t-structure in the derived category of a Grothendieck category associated to a torsion pair in the latter. We revisit the HRS tilting process deriving from it a lot of information on the HRS t-structures which have a projective generator or an injective cogenerator, and obtain several bijections between classes of pairs $(mathcal{A},mathbf{t})$ consisting of an abelian category and a torsion pair in it. We use these bijections to re-prove, by different methods, a recent result of Tilting Theory and the fact that if $mathbf{t}=(mathcal{T},mathcal{F})$ is a torsion pair in a Grothendieck category $mathcal{G}$, then the heart of the associated HRS t-structure is itself a Grothendieck category if, and only if, $mathbf{t}$ is of finite type. We survey this last problem and recent results after its solution.
{"title":"The HRS tilting process and Grothendieck\u0000 hearts of t-structures","authors":"C. Parra, Manuel Saor'in","doi":"10.1090/CONM/769/15416","DOIUrl":"https://doi.org/10.1090/CONM/769/15416","url":null,"abstract":"In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smalo(HSR) t-structure in the derived category of a Grothendieck category associated to a torsion pair in the latter. We revisit the HRS tilting process deriving from it a lot of information on the HRS t-structures which have a projective generator or an injective cogenerator, and obtain several bijections between classes of pairs $(mathcal{A},mathbf{t})$ consisting of an abelian category and a torsion pair in it. We use these bijections to re-prove, by different methods, a recent result of Tilting Theory and the fact that if $mathbf{t}=(mathcal{T},mathcal{F})$ is a torsion pair in a Grothendieck category $mathcal{G}$, then the heart of the associated HRS t-structure is itself a Grothendieck category if, and only if, $mathbf{t}$ is of finite type. We survey this last problem and recent results after its solution.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121565872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-17DOI: 10.2140/pjm.2020.308.473
A. Zimmermann
Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative projectivity, respectively relative injectivity with respect to pairs of adjoint functors. We then modify Auslander Kleiner's construction such that the correspondence holds in the setting of triangulated categories.
{"title":"Green correspondence and relative projectivity for pairs of adjoint functors between triangulated categories","authors":"A. Zimmermann","doi":"10.2140/pjm.2020.308.473","DOIUrl":"https://doi.org/10.2140/pjm.2020.308.473","url":null,"abstract":"Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative projectivity, respectively relative injectivity with respect to pairs of adjoint functors. We then modify Auslander Kleiner's construction such that the correspondence holds in the setting of triangulated categories.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134217156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the well-definedness of some deformations of the fibred biset category in characteristic zero. The method is to realize the fibred biset category and the deformations as the invariant parts of some categories whose compositions are given by simpler formulas. Those larger categories are constructed from a partial category of subcharacters by linearizing and introducing a cocycle.
{"title":"Some deformations of the fibred biset category","authors":"Laurence Barker, İsmail Alperen Öğüt","doi":"10.3906/mat-2001-52","DOIUrl":"https://doi.org/10.3906/mat-2001-52","url":null,"abstract":"We prove the well-definedness of some deformations of the fibred biset category in characteristic zero. The method is to realize the fibred biset category and the deformations as the invariant parts of some categories whose compositions are given by simpler formulas. Those larger categories are constructed from a partial category of subcharacters by linearizing and introducing a cocycle.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114872300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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