{"title":"Local theta correspondence and nilpotent\u0000 invariants","authors":"Chengru Zhu","doi":"10.1090/pspum/101/15","DOIUrl":"https://doi.org/10.1090/pspum/101/15","url":null,"abstract":"We consider two types of nilpotent invariants associated to smooth representations, namely generalized Whittaker models, and associated characters (in the case of a real reductive group). We survey some recent results on the behavior of these nilpotent invariants under local theta correspondence, and highlight the special role of a certain double fiberation of moment maps.","PeriodicalId":237232,"journal":{"name":"Representations of Reductive Groups","volume":"2015 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128596190","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}
{"title":"Explicit local Jacquet-Langlands\u0000 correspondence: The non-dyadic wild case","authors":"C. Bushnell, G. Henniart","doi":"10.1090/pspum/101/03","DOIUrl":"https://doi.org/10.1090/pspum/101/03","url":null,"abstract":"Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $pneq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $text{rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.","PeriodicalId":237232,"journal":{"name":"Representations of Reductive Groups","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129906099","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}
{"title":"On the Casselman-Jacquet functor","authors":"Tsao-Hsien Chen, D. Gaitsgory, Alexander Yom Din","doi":"10.1090/PSPUM/101/04","DOIUrl":"https://doi.org/10.1090/PSPUM/101/04","url":null,"abstract":"We study the Casselman-Jacquet functor $J$, viewed as a functor from the (derived) category of $(mathfrak{g},K)$-modules to the (derived) category of $(mathfrak{g},N^-)$-modules, $N^-$ is the negative maximal unipotent. We give a functorial definition of $J$ as a certain right adjoint functor, and identify it as a composition of two averaging functors $text{Av}^{N^-}_!circ text{Av}^N_*$. We show that it is also isomorphic to the composition $text{Av}^{N^-}_*circ text{Av}^N_!$. Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) $D$-modules on an algebraic stack.","PeriodicalId":237232,"journal":{"name":"Representations of Reductive Groups","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129200250","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}
{"title":"Limiting cycles and periods of Maass\u0000 forms","authors":"A. Reznikov","doi":"10.1090/pspum/101/13","DOIUrl":"https://doi.org/10.1090/pspum/101/13","url":null,"abstract":"We consider (generalized) periods of Maass forms along non-closed geodesics having a closed geodesic as the limit set.","PeriodicalId":237232,"journal":{"name":"Representations of Reductive Groups","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116775429","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}
{"title":"On the generalized Springer\u0000 correspondence","authors":"G. Lusztig","doi":"10.1090/pspum/101/10","DOIUrl":"https://doi.org/10.1090/pspum/101/10","url":null,"abstract":"We close a gap in the explicit determination of the generalized Springer correspondence for a connected reductive group in good characteristic.","PeriodicalId":237232,"journal":{"name":"Representations of Reductive Groups","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133736270","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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