  We extend the classical construction of operator colligations and
characteristic functions. Consider the group $G$ of finite block unitary
matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup
$K=U(\infty)$, which consists of block diagonal unitary matrices (with a block
1 of size $\alpha$ and a matrix $u\in U(\infty)$ repeated $k$ times). It
appears that there is a natural multiplication on the conjugacy classes $G//K$.
We construct 'spectral data' of conjugacy classes, which visualize the
multiplication and are sufficient for a reconstruction of a conjugacy class.
