Compute the isotropy representation

Question

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations.

The isotropy group is $U(1),$ since it stabilizes the point $0.$

I am trying to find explicitly the isotropy representation $\rho: U(1)\rightarrow T_{0} \mathbb{D}\cong \mathbb{R}^2 .$

Any help would be greatly appreciated!!

Answer 1

Let $g=diag(u,u^{-1})\in U(1).$ Then $g$ acts on the unit disc by Möbius tranformations, $z\mapsto u^2z$. This is linear, so the derivative at $z=0$ is the same. Hence the isotropy representation is the square of the standard 1-dim representation of $U(1).$