Let $\pi$ is a standard representation of G = SU(2) on $\mathbb{C}$$^2$. For every z $\in$ $\mathbb{S}$$^1$, define an irreducible unitary representation $\pi$$_z$ of the loop group LG on $\mathbb{C}$$^2$ by
$\pi$$_z$ (f) = $\pi$(f(z)), f $\in$LG
Show that $\pi$$_z$ and $\pi$$_1$ are not unitarily equivalent for z $\neq$ 1.
This question is from the book: Kazhdan's Property (T), Ex4.4.6, there is a hint to consider their characters.
I do not know much about loop groups, the loop group of G is denoted by LG, it is the continous maps from $\mathbb{S}$$^1$ to G, maybe not keep the identity element?