A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of pp-adic groups. The input for this construction is a pair (λ,χ)(lambda , chi ) where λlambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and χchi is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet.
Reeder-Yu 的一篇论文[J. Amer. Math. Soc. 27 (2014), pp.这个构造的输入是一对 ( λ , χ ) (lambda , chi ) ，其中 λ lambda 是来自 Moy-Prasad 滤波的某个表示中的稳定向量，而 χ chi 是残差域的加法群的一个特征。如果得到的超pidal 表示是同构的，我们就说这两对表示是等价的。在本文中，我们描述了这类对的等价类。作为应用，我们给出了分裂邻接群的简单超pidal 表示的分类。最后，在无克拉姆基变化的假设下，我们描述了与这些简单超pidals 相关的朗兰兹参数的性质，证明它们具有微不足道的 L 函数和最小斯旺导体，并证明这些简单超pidals 中的每一个都位于一个单子 L 包中。
{"title":"On input and Langlands parameters for epipelagic representations","authors":"Beth Romano","doi":"10.1090/ert/668","DOIUrl":"https://doi.org/10.1090/ert/668","url":null,"abstract":"<p>A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic groups. The input for this construction is a pair <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis lamda comma chi right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ<!-- χ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(lambda , chi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\"> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a stable vector in a certain representation coming from a Moy–Prasad filtration, and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi\"> <mml:semantics> <mml:mi>χ<!-- χ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet.</p>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"39 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Langlands [On the classification of irreducible representations of real algebraic groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined LL-packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically.
Langlands [On the classification of irreducible representations of real algebraic groups, Math.Surveys Monogr.Math.Soc., Providence, RI, 1989, pp.为了完善局部朗兰兹对应关系，亚当斯-巴尔巴什-沃根 [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc.在回火设置中，卡莱塔[Ann. of Math. (2) 184 (2016), pp.我们证明，Adams-Barbasch-Vogan 和 Kaletha 的回火 L-packets 是相同的，并且参数相同。
{"title":"L-packets over strong real forms","authors":"N. Arancibia Robert, P. Mezo","doi":"10.1090/ert/667","DOIUrl":"https://doi.org/10.1090/ert/667","url":null,"abstract":"<p>Langlands [<italic>On the classification of irreducible representations of real algebraic groups</italic>, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [<italic>The Langlands classification and irreducible characters for real reductive groups</italic>, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically.</p>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"24 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:Let $n$ and $n’$ be positive integers such that $n-n’in {0,1}$. Let $F$ be either $mathbb {R}$ or $mathbb {C}$. Let $K_n$ and $K_{n’}$ be maximal compact subgroups of $mathrm {GL}(n,F)$ and $mathrm {GL}(n’,F)$, respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal $K_n$- and $K_{n’}$-types for pairs of principal series representations of $mathrm {GL}(n,F)$ and $mathrm {GL}(n’,F)$, using their recurrence relations. Our results for $F=mathbb {C}$ can be applied to the arithmetic study of critical values of automorphic $L$-functions.
{"title":"Calculus of archimedean Rankin–Selberg integrals with recurrence relations","authors":"Taku Ishii, Tadashi Miyazaki","doi":"10.1090/ert/618","DOIUrl":"https://doi.org/10.1090/ert/618","url":null,"abstract":"Abstract:Let $n$ and $n’$ be positive integers such that $n-n’in {0,1}$. Let $F$ be either $mathbb {R}$ or $mathbb {C}$. Let $K_n$ and $K_{n’}$ be maximal compact subgroups of $mathrm {GL}(n,F)$ and $mathrm {GL}(n’,F)$, respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal $K_n$- and $K_{n’}$-types for pairs of principal series representations of $mathrm {GL}(n,F)$ and $mathrm {GL}(n’,F)$, using their recurrence relations. Our results for $F=mathbb {C}$ can be applied to the arithmetic study of critical values of automorphic $L$-functions. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"365 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:The Bonnafé–Rouquier equivalence can be seen as a modular analogue of Lusztig’s Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties.
{"title":"Derived equivalences and equivariant Jordan decomposition","authors":"Lucas Ruhstorfer","doi":"10.1090/ert/605","DOIUrl":"https://doi.org/10.1090/ert/605","url":null,"abstract":"Abstract:The Bonnafé–Rouquier equivalence can be seen as a modular analogue of Lusztig’s Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"367 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the branching problem of the metaplectic representation of S p ( 2 , R ) Sp(2, mathbb R) under its principle subgroup S L ( 2 , R ) SL(2, mathbb R) . We find the complete decomposition.
{"title":"Branching of metaplectic representation of 𝑆𝑝(2,ℝ) under its principal 𝕊𝕃(2,ℝ)-subgroup","authors":"GenKai Zhang","doi":"10.1090/ert/609","DOIUrl":"https://doi.org/10.1090/ert/609","url":null,"abstract":"We study the branching problem of the metaplectic representation of S p ( 2 , R ) Sp(2, mathbb R) under its principle subgroup S L ( 2 , R ) SL(2, mathbb R) . We find the complete decomposition.","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"369 1","pages":"498-514"},"PeriodicalIF":0.6,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${mathscr {D}}$-modules on $G$ and $L$, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where $L=T$ is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra ${mathscr {D}}$-module on ${Gtimes T}$. We show that this module is flat over ${mathscr {D}}(T)$, which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of ${mathscr {D}}$-modules.
{"title":"Parabolic induction and the Harish-Chandra 𝒟-module","authors":"Victor Ginzburg","doi":"10.1090/ert/603","DOIUrl":"https://doi.org/10.1090/ert/603","url":null,"abstract":"Abstract:Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${mathscr {D}}$-modules on $G$ and $L$, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where $L=T$ is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra ${mathscr {D}}$-module on ${Gtimes T}$. We show that this module is flat over ${mathscr {D}}(T)$, which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of ${mathscr {D}}$-modules. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"373 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Karin Erdmann, Chrysostomos Psaroudakis, Øyvind Solberg
Abstract:In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.
{"title":"Homological invariants of the arrow removal operation","authors":"Karin Erdmann, Chrysostomos Psaroudakis, Øyvind Solberg","doi":"10.1090/ert/606","DOIUrl":"https://doi.org/10.1090/ert/606","url":null,"abstract":"Abstract:In this paper we show that Gorensteinness, singularity categories and the finite generation condition <span>Fg</span> for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":51304,"journal":{"name":"Representation Theory","volume":"374 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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