2026-09-19T09:29:26Zhttps://dspace.mit.edu/server/oai/request

oai:dspace.mit.edu:1721.1/1177782026-06-16T18:55:46Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022

Pavel Etingof and Ivan Losev. Shelley-Abrahamson, Seth Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics 2018-09-17T14:49:13Z 2018-09-17T14:49:13Z 2018 2018 http://hdl.handle.net/1721.1/117778 1051189673 Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. Cataloged from student-submitted PDF version of thesis. Includes bibliographical references (pages 197-201). This thesis introduces and studies two constructions related to the representation theory of rational Cherednik algebras: the refined filtration by supports for the category O and the Dunkl weight function. The refined filtration by supports provides an analogue for rational Cherednik algebras of the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. In particular, irreducible representations in the rational Cherednik category O with particular generalized support conditions correspond to irreducible representations of associated generalized Hecke algebras. An explicit presentation for these generalized Hecke algebras is given in the Coxeter case, classifying the irreducible finite-dimensional representations in many new cases. The Dunkl weight function K is a holomorphic family of tempered distributions on the real reflection representation of a finite Coxeter group W with values in linear endomorphisms of an irreducible representation of W. The distribution K gives rise to an integral formula for the Gaussian inner product on a Verma module in the rational Cherednik category O. At real parameter values, the restriction of K to the regular locus in the real reflection representation can be interpreted as an analytic function taking values in Hermitian forms, invariant under the braid group, on the image of a Verma module under the Knizhnik-Zamolodchikov (KZ) functor. This provides a bridge between the study of invariant Hermitian forms on representations of rational Cherednik algebras and of Hecke algebras, allowing for a proof that the KZ functor preserves signatures in an appropriate sense. by Seth Shelley-Abrahamson. Ph.D. 201 pages eng Massachusetts Institute of Technology MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582 Mathematics. On representations of rational Cherednik algebras Thesis application/pdf false Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="5913d720-eee2-42d0-80ae-70c13e88a126"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>eng</Language> <Title>On representations of rational Cherednik algebras</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2018</PublicationDate> <Authors> <Author> <DisplayName>Shelley-Abrahamson, Seth</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://dspace.mit.edu/handle/1721.1/7582</License> <Keyword>Mathematics.</Keyword> <Abstract>This thesis introduces and studies two constructions related to the representation theory of rational Cherednik algebras: the refined filtration by supports for the category O and the Dunkl weight function. The refined filtration by supports provides an analogue for rational Cherednik algebras of the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. In particular, irreducible representations in the rational Cherednik category O with particular generalized support conditions correspond to irreducible representations of associated generalized Hecke algebras. An explicit presentation for these generalized Hecke algebras is given in the Coxeter case, classifying the irreducible finite-dimensional representations in many new cases. The Dunkl weight function K is a holomorphic family of tempered distributions on the real reflection representation of a finite Coxeter group W with values in linear endomorphisms of an irreducible representation of W. The distribution K gives rise to an integral formula for the Gaussian inner product on a Verma module in the rational Cherednik category O. At real parameter values, the restriction of K to the regular locus in the real reflection representation can be interpreted as an analytic function taking values in Hermitian forms, invariant under the braid group, on the image of a Verma module under the Knizhnik-Zamolodchikov (KZ) functor. This provides a bridge between the study of invariant Hermitian forms on representations of rational Cherednik algebras and of Hecke algebras, allowing for a proof that the KZ functor preserves signatures in an appropriate sense.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>