  This paper is devoted to a Stackelberg stochastic differential game for a
linear mean-field type stochastic differential system with a mean-field type
quadratic cost functional in finite horizon. The coefficients in the state
equation and weighting matrices in the cost functional are all deterministic.
Closed-loop Stackelberg equilibrium strategies are introduced which require to
be independent of initial states. Follower's problem is solved firstly, which
is a stochastic linear quadratic optimal control problem. By converting the
original problem into a new one whose optimal control is known, the closed-loop
optimal strategy of the follower is characterized by two coupled Riccati
equations as well as a linear mean-field type backward stochastic differential
equation. Then the leader turns to solve a stochastic linear quadratic optimal
control problem for a mean-field type forward-backward stochastic differential
equation. Necessary conditions for the existence of closed-loop optimal
strategies for the leader is given by the existence of two coupled Riccati
equations with a linear mean-field type backward stochastic differential
equation. The solvability of Riccati equations of leader's optimization problem
is discussed in the case where the diffusion term of the state equation does
not contain the control process of the follower. Moreover, leader's value
function is expressed via two backward stochastic differential equations and
two Lyapunov equations.
