  In this paper, we mainly study the additional symmetry and $\tau$ functions
of a multi-component Gelfand-Dickey hierarchy which includes many classical
integrable systems, such as the multi-component KdV hierarchy and the
multi-component Boussinesq hierarchy. With other kinds of reductions, we can
derive a B type multi-component Gelfand-Dickey hierarchy and a C type
multi-component Gelfand-Dickey hierarchy. In our research, the additional flows
of the additional symmetries can not all survive. By calculating, we find that
the generator of the additional symmetry of the C type multi-component
Gelfand-Dickey hierarchy is different from that of the B type multi-component
Gelfand-Dickey hierarchy, while the forms of their surviving additional flows
are the same.
