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

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.