Weil Representations of Unitary Groups

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.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Doctoral -- first

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Faculty of Graduate Studies and Research, University of Regina

Year dc:date.issued

2019

Author dc:creator

• Shau, Moumita

Advisors dc:contributor.advisor

• Herman, Allen
• Szechtman, Fernando

Committee members dc:contributor.committeemember

• Sadaoui-Mouhoub, Samira
• Stanley, Donald
• Meagher, Karen

Language dc:language.iso

en

DOI

https://doi.org/10.82465/3684

OAI identifier oai:identifier

oai:uregina.scholaris.ca:10294/9181