Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

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.