On the unitary representations of the affine $ax+b$-group, $\widehat{sl}(2,\mathbb{R})$ and their relatives

TL;DR

This paper surveys the construction and analysis of unitary representations of the loop $ax+b$-group and the affine $ ext{sl}(2, ext{R})$-algebra, highlighting their mathematical structures and physical motivations.

Contribution

It consolidates previous results on unitary representations and introduces related concepts like the loop $ ext{Gamma}$-function and continuous series for $ ext{sl}(2, ext{R})$.

Findings

Connection between loop $ax+b$-group representations and loop $ ext{Gamma}$-function.

Construction of continuous series for $ ext{sl}(2, ext{R})$-algebra.

Survey of recent results with physical and mathematical motivations.

Abstract

This article focuses on two related topics: unitary representations of the loop $ax+b$-group and their relation to a loop version of the $\Gamma$-function and the construction of continuous series for the $\widehat{sl(2,\mathbb{R})}$-algebra. Mainly this is a survey of some results from arXiv:1012.4826 , arXiv:1210.2135 alongside with the motivation for them both from the physical and mathematical points of view.