Decomposition of irreducible unitary representations of the group $SL(2C)$ restricted to the subgroup $SU(1,1)$. The complementary series

A study is made of the decomposition of the irreducible unitary representations of the complementary series of the group $SL(2C)$ in the case when they are restricted to the subgroup $SU(1,1)$ into irreducible representations of the group $SU(1,1)$. Calculations are made of the matrix elements of the representations of the complementary series of the group $SL(2C)$. A formula is obtained that connects thematrix elements of the operators of the representations of $SL(2C)$ and $SU(1,1)$; it is analytically continued with respect to the parameter $\sigma$, which characterizes the representation of the complementary series. Since the singularities of the integrand in this formula are poles of second order, the expression obtained by analytic continuation contains not only the matrix elements of the operators of the representations of $SU(1,1)$, as in the case of the representations of the principal series, but also their derivatives with respect to the parameter $l$, which characterizes the representation of the group $SU(1,1)$.