A classification is a list of objects grouped by certain properties. In Science,
classifications play a crucial role. Indeed, for researchers seeking an object with
particular traits, referring to these classifications enables them to identify potential
objects they`re dealing with.
The local Langlands program stands as one of the most significant advancements in
Number Theory in recent decades. This program is conjectured to classify the set of
representations of certain local groups in terms of some arithmetic parameters. These
conjectures were recently proved by Jim Arthur (using the collective work of many
mathematicians) for some groups which are called classical, and which hold
significant importance beyond mathematics, extending into fields like Physics.
Nevertheless, this classification remains highly abstract: although each arithmetic
parameter corresponds to a unique representation, characterizing the properties of
the representation based solely on the parameter remains a challenging task.
In this project we will try to understand how the parameter looks like in the case the
representation is unitarizable: these representations are of great importance in
various scientific domains, making it a considerable feat to classify them.
Since Number Theory often draws from practical experience, this research project will
commence by establishing simple examples. These will serve as foundational
illustrations to illuminate the potential construction of the overarching theory.
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