{"id":{"repo_id":"regina","oai_identifier":"oai:uregina.scholaris.ca:10294/9181"},"canonical_url":"https://search.dev.ndltd.org/etd/regina/oai:uregina.scholaris.ca:10294/9181","repository":{"repo_id":"regina","name":"University of Regina","base_url":"https://uregina.scholaris.ca/server/oai/request"},"display":{"title":"Weil Representations of Unitary Groups","abstract":"Let A be a finite, commutative, local and principal ideal ring, with maximal ideal r, endowed with an involution , possibly trivial, and let R be the fixed ring of . We distinguish three types of extensions A=R: symplectic, when is trivial and A = R; unramified, when the involution that induces on A=r is nontrivial; ramified, when the involution that induces on A=r is trivial. The last case further divides into two cases, even or odd, depending on the nilpotency degree of r. Whatever the case, let Um(A) be the unitary group of rank m over A associated to a nondegenerate hermitian or skew-hermitian form h relative to . Specifically, in order to obtain an imbedding of Um(A) into a suitable symplectic group over R, we take h to be skew-hermitian in all cases except in the ramified even case. We are interested in the decomposition of the Weil representation of unitary group Um(A) into irreducible constituents and a Cli ord theory description of these constituents. This decomposition was achieved in [CMS1], [S] and [HS] in the symplectic, unramified and ramified even cases, respectively. We solve the decomposition problem in the last remaining case, namely the ramified odd case. Moreover, we show that this Weil representation is multiplicity free. Furthermore, restriction to the special unitary group SU2n(A) preserves irreducibility and multiplicity freeness, provided n > 1. On the other hand, a Cli ord theory description of all irreducible constituents of the Weil module was given in [CMS2] in the symplectic case. Using di erent techniques, we achieve a uniform Cli ord theory description of all Weil components in all four cases described above, with respect to all abelian congruence subgroups ofUm(A) and a third of its nonabelian congruence subgroups.","abstract_html":"Let A be a finite, commutative, local and principal ideal ring, with maximal ideal r, endowed with an involution , possibly trivial, and let R be the fixed ring of . We distinguish three types of extensions A=R: symplectic, when is trivial and A = R; unramified, when the involution that induces on A=r is nontrivial; ramified, when the involution that induces on A=r is trivial. The last case further divides into two cases, even or odd, depending on the nilpotency degree of r. Whatever the case, let Um(A) be the unitary group of rank m over A associated to a nondegenerate hermitian or skew-hermitian form h relative to . Specifically, in order to obtain an imbedding of Um(A) into a suitable symplectic group over R, we take h to be skew-hermitian in all cases except in the ramified even case. We are interested in the decomposition of the Weil representation of unitary group Um(A) into irreducible constituents and a Cli ord theory description of these constituents. This decomposition was achieved in [CMS1], [S] and [HS] in the symplectic, unramified and ramified even cases, respectively. We solve the decomposition problem in the last remaining case, namely the ramified odd case. Moreover, we show that this Weil representation is multiplicity free. Furthermore, restriction to the special unitary group SU2n(A) preserves irreducibility and multiplicity freeness, provided n > 1. On the other hand, a Cli ord theory description of all irreducible constituents of the Weil module was given in [CMS2] in the symplectic case. Using di erent techniques, we achieve a uniform Cli ord theory description of all Weil components in all four cases described above, with respect to all abelian congruence subgroups ofUm(A) and a third of its nonabelian congruence subgroups.","abstract_has_math":false,"creators":["Shau, Moumita"],"institution":"Faculty of Graduate Studies and Research, University of Regina","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral -- first","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Herman, Allen","Szechtman, Fernando"],"committee_chairs":[],"committee_members":["Sadaoui-Mouhoub, Samira","Stanley, Donald","Meagher, Karen"],"year":2019,"date_issued":"2019-08","date_published":"2019-08","updated_at":"2026-07-24T04:03:25Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/3684"],"render_values":[{"text":"https://doi.org/10.82465/3684","href":"https://doi.org/10.82465/3684","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10294/9181","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Herman, Allen","Szechtman, Fernando"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Sadaoui-Mouhoub, Samira","Stanley, Donald","Meagher, Karen"]},{"key":"dc:creator","label":"Author","values":["Shau, Moumita"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-08-26T22:29:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-08-26T22:29:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-08"]},{"key":"dc:publisher","label":"Institution","values":["Faculty of Graduate Studies and Research, University of Regina"]},{"key":"dc:type","label":"Dc Type","values":["master thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral -- first"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Faculty of Graduate Studies and Research, University of Regina"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/3684"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10294/9181"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics, University of Regina. vii, 99 p."]},{"key":"dc:description.abstract","label":"Abstract","values":["Let A be a finite, commutative, local and principal ideal ring, with maximal ideal r, endowed with an involution , possibly trivial, and let R be the fixed ring of . We distinguish three types of extensions A=R: symplectic, when is trivial and A = R; unramified, when the involution that induces on A=r is nontrivial; ramified, when the involution that induces on A=r is trivial. The last case further divides into two cases, even or odd, depending on the nilpotency degree of r. Whatever the case, let Um(A) be the unitary group of rank m over A associated to a nondegenerate hermitian or skew-hermitian form h relative to . Specifically, in order to obtain an imbedding of Um(A) into a suitable symplectic group over R, we take h to be skew-hermitian in all cases except in the ramified even case. We are interested in the decomposition of the Weil representation of unitary group Um(A) into irreducible constituents and a Cli ord theory description of these constituents. This decomposition was achieved in [CMS1], [S] and [HS] in the symplectic, unramified and ramified even cases, respectively. We solve the decomposition problem in the last remaining case, namely the ramified odd case. Moreover, we show that this Weil representation is multiplicity free. Furthermore, restriction to the special unitary group SU2n(A) preserves irreducibility and multiplicity freeness, provided n > 1. On the other hand, a Cli ord theory description of all irreducible constituents of the Weil module was given in [CMS2] in the symplectic case. Using di erent techniques, we achieve a uniform Cli ord theory description of all Weil components in all four cases described above, with respect to all abelian congruence subgroups ofUm(A) and a third of its nonabelian congruence subgroups."]},{"key":"dc:title","label":"Title","values":["Weil Representations of Unitary Groups"]}]}],"canonical_facts":{"dc:contributor.advisor":["Herman, Allen","Szechtman, Fernando"],"dc:contributor.committeemember":["Sadaoui-Mouhoub, Samira","Stanley, Donald","Meagher, Karen"],"dc:creator":["Shau, Moumita"],"dc:date.accessioned":["2020-08-26T22:29:28Z"],"dc:date.available":["2020-08-26T22:29:28Z"],"dc:date.issued":["2019-08"],"dc:description":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics, University of Regina. vii, 99 p."],"dc:description.abstract":["Let A be a finite, commutative, local and principal ideal ring, with maximal ideal r, endowed with an involution , possibly trivial, and let R be the fixed ring of . We distinguish three types of extensions A=R: symplectic, when is trivial and A = R; unramified, when the involution that induces on A=r is nontrivial; ramified, when the involution that induces on A=r is trivial. The last case further divides into two cases, even or odd, depending on the nilpotency degree of r. Whatever the case, let Um(A) be the unitary group of rank m over A associated to a nondegenerate hermitian or skew-hermitian form h relative to . Specifically, in order to obtain an imbedding of Um(A) into a suitable symplectic group over R, we take h to be skew-hermitian in all cases except in the ramified even case. We are interested in the decomposition of the Weil representation of unitary group Um(A) into irreducible constituents and a Cli ord theory description of these constituents. This decomposition was achieved in [CMS1], [S] and [HS] in the symplectic, unramified and ramified even cases, respectively. We solve the decomposition problem in the last remaining case, namely the ramified odd case. Moreover, we show that this Weil representation is multiplicity free. Furthermore, restriction to the special unitary group SU2n(A) preserves irreducibility and multiplicity freeness, provided n > 1. On the other hand, a Cli ord theory description of all irreducible constituents of the Weil module was given in [CMS2] in the symplectic case. Using di erent techniques, we achieve a uniform Cli ord theory description of all Weil components in all four cases described above, with respect to all abelian congruence subgroups ofUm(A) and a third of its nonabelian congruence subgroups."],"dc:identifier.doi":["https://doi.org/10.82465/3684"],"dc:identifier.uri":["https://hdl.handle.net/10294/9181"],"dc:language.iso":["en"],"dc:publisher":["Faculty of Graduate Studies and Research, University of Regina"],"dc:title":["Weil Representations of Unitary Groups"],"dc:type":["master thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral -- first"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Faculty of Graduate Studies and Research, University of Regina"]},"updated_at":"2026-07-24T04:03:25Z"}