ИСТИНА

For numerous infinite dimensional (’large’) groups there exist natural operations of multiplication of double cosets or conjugacy class. We discuss real classical groups. For instance, denote by G = U(∞) the group of infinite unitary matrices g such that g − 1 have only finite number of matrix elements, by Kn - the stabilizer of the first n basis vectors. Consider the product G × · · · × G (α times) and consider conjugacy classes of this group with respect to the diagonal subgroup Kn. We define multiplication on conjugacy classes and explain how to visualize it using appropriate ’spectral data’.