Analytic continuation of functions defined on subgroups of the complex Lorentz group

A question that arises in the group-theoretical approach to the problem of conspiring Regge trajectories is discussed - the analytic continuation of functions defined on subgroups of the complex Lorentz group. It is shown that a real-analytic function $f(\varphi,\cos\theta,\psi)$ on $SU(2)$ that is analytic in $\omega=\cos\theta$ in the whole plane can be continued to a complex-analytic function on $SL(2C)$.