  We consider a class of linear-quadratic-Gaussian mean-field games with a
major agent and considerable heterogeneous minor agents in the presence of
mean-field interactions. The individual admissible controls are constrained in
closed convex subsets $\Gamma_{k}$ of $\mathbb{R}^{m}.$ The decentralized
strategies for individual agents and consistency condition system are
represented in an unified manner through a class of mean-field forward-backward
stochastic differential equations involving projection operators on
$\Gamma_{k}$. The well-posedness of consistency system is established in both
the local and global cases by the contraction mapping and discounting method
respectively. Related $\varepsilon-$Nash equilibrium property is also verified.
