Unitary Irreducible Highest Weight Modules of su(2,1) of Highest Weight λ - ρ Where λ is Integral Regular

In this thesis, the following problem is tackled: for the real form su(2, 1) of sl3(C), compute chsL(x(−λ)) for λ ∈ Λ+ regular (i.e., λ integral, dominant, regular weights) and x ∈ W, and determine when the representations are unitary (i.e. when chsL(x(−λ)) = chL(x(−λ))). Here, L(x(−λ)) is the unique irreducible quotient of the Verma module M(x(−λ)) of highest weight x(−λ) − ρ. One of the most important open problems in Mathematics is called The Unitary Dual Problem: classify the irreducible unitary representations of a group. This was introduced in the 1930s by Israel Gelfand. In the realm of unitary representations, the initial commencement often involves portraying the Hermitian representations, those that harbor an invariant Hermitian form. Following this classification, one embarks on calculating the signatures of these invariant forms within the Hermitian tapestry. The final act in this symphony of classification involves discerning which of these forms embrace positivity or negativity in their definiteness. For the problem at hand, we heavily rely on the machineries developed particularly in [Yee19]. For the enthusiast, it is recommended to explore the arcane landscapes the author had ventured into. For our purposes, we explicitly deal with computations. Much of the chapters included are self-contained. Since the key players in this Mathematical fabula scaenica are “weight spaces”, “Weyl groups” and “affine Weyl groups”, their masquerading nature had been unveiled through elucidation. Whenever new notations had risen, the meaning associated with it had been expounded on. For now, we let the sea advance in silence.

Name thesis:degree_name

M.Sc.

Discipline thesis:degree_discipline

Mathematics and Statistics

Grantor

University of Windsor

Year dc:date.issued

2024

Author dc:creator

• Mobassir, Yasin

Advisor dc:contributor.advisor

• Wai Ling Yee

Contributors dc:contributor

• scholarship@uwindsor.ca

Licence

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language.iso

en_CA

Handle dc:identifier.uri

https://hdl.handle.net/20.500.14776/8462

OAI identifier oai:identifier

oai:uwindsor.scholaris.ca:20.500.14776/8462