  This paper investigates a linear quadratic mean field leader-follower team
problem, where the model involves one leader and a large number of
weakly-coupled interactive followers. The leader and the followers cooperate to
optimize the social cost. Specifically, for any strategy provided first by the
leader, the followers would like to choose a strategy to minimize social cost
functional. Using variational analysis and person-by-person optimality, we
construct two auxiliary control problems. By solving sequentially the auxiliary
control problems with consistent mean field approximations, we can obtain a set
of decentralized social optimality strategy with help of a class of
forward-backward consistency systems. The relevant Stackelberg equilibrium is
further proved under some proper conditions.
