The problem about classification of characters of the countable groups was solved for wide family of the groups: for infinite symmetric group (E.Thoma's theorem) etc. The realization of the corresponding representations was done later and it happened the question reduces to analysis of the adjoint action of the group on the lattice of all its subgroups. The question about the list adjoint-invariant measures (consquently list of chatracters) - on those lattices for various groups will be discussed.