{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8672"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8672","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Orbital varieties and unipotent representations of classical semisimple Lie group","abstract":"Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein's parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. The parameter set is the family of standard Young or domino tableau of a given shape. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. This parameterization is particularly useful; it provides a method of determining the r-invariant of each orbital variety, as well as a way of relating an orbital variety in any classical group to one lying in type A.","abstract_html":"Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein's parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. The parameter set is the family of standard Young or domino tableau of a given shape. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. This parameterization is particularly useful; it provides a method of determining the r-invariant of each orbital variety, as well as a way of relating an orbital variety in any classical group to one lying in type A.","abstract_has_math":false,"creators":["Pietraho, Thomas, 1973-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["David A. Vogan."],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001","date_published":"2001","updated_at":"2026-07-22T22:21:55Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/8672","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["David A. Vogan."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Pietraho, Thomas, 1973-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2005-08-23T22:12:29Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2005-08-23T22:12:29Z"]},{"key":"dc:date.issued","label":"Date","values":["2001"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/8672"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.","Includes bibliographical references (p. 81-83)."]},{"key":"dc:description.abstract","label":"Abstract","values":["Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein's parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. The parameter set is the family of standard Young or domino tableau of a given shape. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. This parameterization is particularly useful; it provides a method of determining the r-invariant of each orbital variety, as well as a way of relating an orbital variety in any classical group to one lying in type A.","(cont.) The second part of the thesis addresses the representations V(V, ir) constructed by Graham and Vogan. A natural question is how well the V(V, 7r) approximate the set of unipotent representations that ought to be attached to the nilpotent orbit 0. The answer is promising in the setting of spherical orbits. When it is possible to carry out the Graham-Vogan construction, the corresponding infinitesimal character lies in the set of characters suggested by W. M. McGovern. Furthermore, we show that it is possible to carry out the Graham-Vogan construction for a suffient number of orbital varieties to account for all the infinitesimal characters attached to 0 by McGovern."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Orbital varieties and unipotent representations of classical semisimple Lie group"]}]}],"canonical_facts":{"dc:contributor.advisor":["David A. Vogan."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Pietraho, Thomas, 1973-"],"dc:date.accessioned":["2005-08-23T22:12:29Z"],"dc:date.available":["2005-08-23T22:12:29Z"],"dc:date.issued":["2001"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.","Includes bibliographical references (p. 81-83)."],"dc:description.abstract":["Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein's parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. The parameter set is the family of standard Young or domino tableau of a given shape. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. This parameterization is particularly useful; it provides a method of determining the r-invariant of each orbital variety, as well as a way of relating an orbital variety in any classical group to one lying in type A.","(cont.) The second part of the thesis addresses the representations V(V, ir) constructed by Graham and Vogan. A natural question is how well the V(V, 7r) approximate the set of unipotent representations that ought to be attached to the nilpotent orbit 0. The answer is promising in the setting of spherical orbits. When it is possible to carry out the Graham-Vogan construction, the corresponding infinitesimal character lies in the set of characters suggested by W. M. McGovern. Furthermore, we show that it is possible to carry out the Graham-Vogan construction for a suffient number of orbital varieties to account for all the infinitesimal characters attached to 0 by McGovern."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1721.1/8672"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Orbital varieties and unipotent representations of classical semisimple Lie group"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:55Z"}