The $W_{1+\infty}(gl_s)$--symmetries of the $S$--component KP hierarchy

Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchyto the $W_{1+\infty}$--, respectively $W_{1+\infty}\times W_{1+\infty}$--algebra symmeties. In this paper we generalize the results to the $s$--component KP hierarchy. The vertex operators generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of $W_{1+\infty}$. Since the Toda lattice hierarchy is equivalent to the $2$--component KP hierarchy, the results of this paper uncover in that particular case a much richer structure than the one obtained by Adler, Shiota and van Moerbeke.