These properties follow from the orthogonality relations for characters of representations of finite groups. For any non-zero in define the representation on the finite-dimensional inner product space by  It follows that By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups , particularly: Actually, all irreducible representations of on which the center acts nontrivially arise in this way. The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups.