How to calculate the group action $ SU (1,1) $ on $ SU (1,1) / U (1) $

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How to calculate the group action $ SU (1,1) $ on $ SU (1,1)/U(1) $?

Where $ SU (1,1)/U(1) $ is group quotient.

I know that the general form of a matrix in SU(1,1) is given by

$(\begin{array}{ccc} \alpha & \beta \\ \beta^* & \alpha^* \end{array})$, where $|\alpha|^2 - |\beta|^2 = 1$

Can you suggest articles? or help me solve it?

Please.