Let $GSpin(V)$ (resp. $GPin(V)$) be a general spin group (resp. a general Pin group) associated with a nondegenerate quadratic space $V$ of dimension $n$ over an Archimedean local field $F$. For a nondegenerate quadratic space $W$ of dimension $n-1$ over $F$, we also consider $GSpin(W)$ and $GPin(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($GSpin(V), GSpin(W)$) and also for a pair of groups ($GPin(V), GPin(W)$); namely, we prove that the restriction to $GSpin(W)$ (resp. $GPin(W)$) of an irreducible Casselman-Wallach representation of $GSpin(V)$ (resp. $GPin(V)$) is multiplicity free.
让$GSpin(V)$ (resp. $GPin(V)$)是一个与阿基米德局部域$F$上维数为$n的非enerate二次元空间$V$相关联的一般自旋群（res. a general Pin group）。对于维数为 $n-1$ over $F$ 的非enerate 二次空间 $W$，我们也考虑 $GSpin(W)$ 和 $GPin(W)$。我们证明了一对群（$GSpin(V), GSpin(W)$）和一对群（$GPin(V), GPin(W)$）在阿基米德情况下的多重性定理；即，我们证明了对 $GSpin(W)$ 的限制（respect.的一个不可还原的卡塞尔曼-瓦拉几表示是无多重性的。
{"title":"Multiplicity One Theorem for General Spin Groups: The Archimedean Case","authors":"Melissa Emory, Yeansu Kim, Ayan Maiti","doi":"arxiv-2409.09320","DOIUrl":"https://doi.org/arxiv-2409.09320","url":null,"abstract":"Let $GSpin(V)$ (resp. $GPin(V)$) be a general spin group (resp. a general\u0000Pin group) associated with a nondegenerate quadratic space $V$ of dimension $n$\u0000over an Archimedean local field $F$. For a nondegenerate quadratic space $W$ of\u0000dimension $n-1$ over $F$, we also consider $GSpin(W)$ and $GPin(W)$. We prove\u0000the multiplicity-at-most-one theorem in the Archimedean case for a pair of\u0000groups ($GSpin(V), GSpin(W)$) and also for a pair of groups ($GPin(V),\u0000GPin(W)$); namely, we prove that the restriction to $GSpin(W)$ (resp.\u0000$GPin(W)$) of an irreducible Casselman-Wallach representation of $GSpin(V)$\u0000(resp. $GPin(V)$) is multiplicity free.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253725","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 introduce two novel techniques that simplify calculation of Jordan-Kronecker invariants for a Lie algebra $mathfrak{g}$ and for a Lie algebra representation $rho$. First, the stratification of matrix pencils under strict equivalence puts restrictions on the Jordan-Kronecker invariants. Second, we show that the Jordan-Kronecker invariants of a semi-direct sum $mathfrak{g} ltimes_{rho} V$ are sometimes determined by the Jordan-Kronecker invariants of the dual Lie algebra representation $rho^*$.
{"title":"New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations","authors":"I. K. Kozlov","doi":"arxiv-2409.09535","DOIUrl":"https://doi.org/arxiv-2409.09535","url":null,"abstract":"We introduce two novel techniques that simplify calculation of\u0000Jordan-Kronecker invariants for a Lie algebra $mathfrak{g}$ and for a Lie\u0000algebra representation $rho$. First, the stratification of matrix pencils\u0000under strict equivalence puts restrictions on the Jordan-Kronecker invariants.\u0000Second, we show that the Jordan-Kronecker invariants of a semi-direct sum\u0000$mathfrak{g} ltimes_{rho} V$ are sometimes determined by the\u0000Jordan-Kronecker invariants of the dual Lie algebra representation $rho^*$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253721","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}
Mrigendra Singh Kushwaha, K. N. Raghavan, Sankaran Viswanath
We consider the affine Lie algebra $widehat{mathfrak{sl}_2}$ and the Kostant-Kumar submodules of tensor products of its level 1 highest weight integrable representations. We construct crystals for these submodules in terms of the charged partitions model and describe their decomposition into irreducibles.
{"title":"Crystals for Kostant-Kumar modules of $widehat{mathfrak{sl}_2}$","authors":"Mrigendra Singh Kushwaha, K. N. Raghavan, Sankaran Viswanath","doi":"arxiv-2409.09328","DOIUrl":"https://doi.org/arxiv-2409.09328","url":null,"abstract":"We consider the affine Lie algebra $widehat{mathfrak{sl}_2}$ and the\u0000Kostant-Kumar submodules of tensor products of its level 1 highest weight\u0000integrable representations. We construct crystals for these submodules in terms\u0000of the charged partitions model and describe their decomposition into\u0000irreducibles.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253723","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}
This paper is a continuation of Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification [1]. In that paper, we found all groups G for which the Terwilliger algebra of the group association scheme, denoted T (G), is almost commutative. We also found the primitive idempotents for T (G) for three of the four types of such groups. In this paper, we determine the primitive idempotents for the fourth type.
本文是群关联模式的几乎交换 Terwilliger Algebras ofGroup Association Schemes I 的继续：分类 [1]。在那篇论文中，我们找到了群关联方案的特威里格代数（表示为 T (G)）几乎是交换的所有群 G。我们还找到了四类群中三类群的 T (G) 的原始empotents。在本文中，我们将确定第四种类型的基元幂等式。
{"title":"Almost Commutative Terwilliger Algebras of Group Association Schemes II: Primitive Idempotents","authors":"Nicholas L. Bastian","doi":"arxiv-2409.09482","DOIUrl":"https://doi.org/arxiv-2409.09482","url":null,"abstract":"This paper is a continuation of Almost Commutative Terwilliger Algebras of\u0000Group Association Schemes I: Classification [1]. In that paper, we found all\u0000groups G for which the Terwilliger algebra of the group association scheme,\u0000denoted T (G), is almost commutative. We also found the primitive idempotents\u0000for T (G) for three of the four types of such groups. In this paper, we\u0000determine the primitive idempotents for the fourth type.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253722","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}
Mogelin-Renard parametrize A-packet of unitary group through cohomological induction in good parity case. Each parameter gives rise to an $A_{mathfrak q}(lambda)$ which is either $0$ or irreducible. Trapa proposed an algorithm to determine whether a mediocre $A_{mathfrak q}(lambda)$ of $mathrm U(p, q)$ is non-zero. Based on his result, we present a further understanding of the non-vanishing condition of Mogelin-Renard's parametrization. Our criterion come out to be a system of linear constraints, and very similiar to the $p$-adic case.
{"title":"Non-vanishing condition on Mogelin-Renard's parametrization for Arthur packets of $mathrm U(p,q)$","authors":"Chang Huang","doi":"arxiv-2409.09358","DOIUrl":"https://doi.org/arxiv-2409.09358","url":null,"abstract":"Mogelin-Renard parametrize A-packet of unitary group through cohomological\u0000induction in good parity case. Each parameter gives rise to an $A_{mathfrak\u0000q}(lambda)$ which is either $0$ or irreducible. Trapa proposed an algorithm to\u0000determine whether a mediocre $A_{mathfrak q}(lambda)$ of $mathrm U(p, q)$ is\u0000non-zero. Based on his result, we present a further understanding of the\u0000non-vanishing condition of Mogelin-Renard's parametrization. Our criterion come\u0000out to be a system of linear constraints, and very similiar to the $p$-adic\u0000case.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"187 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253724","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}
For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.
{"title":"The stack of spherical Langlands parameters","authors":"Thibaud van den Hove","doi":"arxiv-2409.09522","DOIUrl":"https://doi.org/arxiv-2409.09522","url":null,"abstract":"For a reductive group over a nonarchimedean local field, we define the stack\u0000of spherical Langlands parameters, using the inertia-invariants of the\u0000Langlands dual group. This generalizes the stack of unramified Langlands\u0000parameters in case the group is unramified. We then use this stack to deduce\u0000the Eichler--Shimura congruence relations for Hodge type Shimura varieties,\u0000without restrictions on the ramification.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253730","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}
Suppose $mathfrak{g}$ is a semisimple complex Lie algebra and $mathfrak{h}$ is a Cartan subalgebra of $mathfrak{g}$. To the pair $(mathfrak{g},mathfrak{h})$ one can associate both a Weyl group and a set of Kac diagrams. There is a natural map from the set of elliptic conjugacy classes in the Weyl group to the set of Kac diagrams. In both this setting and the twisted setting, this paper (a) shows that this map is injective and (b) explicitly describes this map's image.
{"title":"Kac Diagrams for Elliptic Weyl Group Elements","authors":"Stephen DeBacker, Jacob Haley","doi":"arxiv-2409.09255","DOIUrl":"https://doi.org/arxiv-2409.09255","url":null,"abstract":"Suppose $mathfrak{g}$ is a semisimple complex Lie algebra and $mathfrak{h}$\u0000is a Cartan subalgebra of $mathfrak{g}$. To the pair\u0000$(mathfrak{g},mathfrak{h})$ one can associate both a Weyl group and a set of\u0000Kac diagrams. There is a natural map from the set of elliptic conjugacy classes\u0000in the Weyl group to the set of Kac diagrams. In both this setting and the\u0000twisted setting, this paper (a) shows that this map is injective and (b)\u0000explicitly describes this map's image.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253726","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}
This survey article, written for the Encyclopedia of Mathematical Physics, 2nd edition, is devoted to the remarkable family of operators introduced by Charles Dunkl and to their $q$-analogues discovered by Ivan Cherednik. The main focus is on the r^ole of these operators in studying integrable many-body systems such as the Calogero-Moser and the Ruijsenaars systems. To put these constructions into a wider context, we indicate their relationship with the theory of the rational Cherednik algebras and double affine Hecke algebras. While we do not include proofs, references to the original research articles are provided, accompanied by brief historical comments.
{"title":"Dunkl and Cherednik operators","authors":"Oleg Chalykh","doi":"arxiv-2409.09005","DOIUrl":"https://doi.org/arxiv-2409.09005","url":null,"abstract":"This survey article, written for the Encyclopedia of Mathematical Physics,\u00002nd edition, is devoted to the remarkable family of operators introduced by\u0000Charles Dunkl and to their $q$-analogues discovered by Ivan Cherednik. The main\u0000focus is on the r^ole of these operators in studying integrable many-body\u0000systems such as the Calogero-Moser and the Ruijsenaars systems. To put these\u0000constructions into a wider context, we indicate their relationship with the\u0000theory of the rational Cherednik algebras and double affine Hecke algebras.\u0000While we do not include proofs, references to the original research articles\u0000are provided, accompanied by brief historical comments.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253737","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 characterize locally finite non-positive dg algebras that arise as Koszul duals of locally finite non-positive dg algebras. Moreover, we show that the Koszul dual functor induces contravariant derived equivalnces. As a consequence, we prove that every functorially finite bounded heart of $pvd A$ of a locally finite non-positive dg algebra is a length category.
{"title":"Contravariant Koszul duality between non-positive and positive dg algebras","authors":"Riku Fushimi","doi":"arxiv-2409.08842","DOIUrl":"https://doi.org/arxiv-2409.08842","url":null,"abstract":"We characterize locally finite non-positive dg algebras that arise as Koszul\u0000duals of locally finite non-positive dg algebras. Moreover, we show that the\u0000Koszul dual functor induces contravariant derived equivalnces. As a\u0000consequence, we prove that every functorially finite bounded heart of $pvd A$\u0000of a locally finite non-positive dg algebra is a length category.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253732","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}
This research aims to define Kac-Moody Lie algebra in Quaternion by using the concept of Quaternification of Lie algebra. The results of this research obtained the definition of Universal Kac-Moody Quaternion Lie algebra, Standard Kac-Moody Quaternion Lie algebra, and Reduced Kac-Moody Quaternion Lie algebra
{"title":"Kac-Moody Quaternion Lie Algebra","authors":"Ferdi, Amir Kamal Amir, Andi Muhammad Anwar","doi":"arxiv-2409.10396","DOIUrl":"https://doi.org/arxiv-2409.10396","url":null,"abstract":"This research aims to define Kac-Moody Lie algebra in Quaternion by using the\u0000concept of Quaternification of Lie algebra. The results of this research\u0000obtained the definition of Universal Kac-Moody Quaternion Lie algebra, Standard\u0000Kac-Moody Quaternion Lie algebra, and Reduced Kac-Moody Quaternion Lie algebra","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253717","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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