  Questions on random matrices and on non-intersecting Brownian motions have
led to the study of moment matrices with regard to several weights. The purpose
of this paper is to show that the determinants of such moment matrices satisfy,
upon adding one set of time deformations for each weight, the multi-component
KP-hierarchy: these determinants are thus "tau-functions" for these integrable
hierarchies. The tau-functions, so obtained, with appropriate shifts of the
time-parameters (forward and backwards) will be expressed in terms of multiple
orthogonal polynomials for these weights and their Cauchy transforms. As an
application, the multi-component KP-hierarchy leads to a large set of
non-linear PDE's, which are useful in finding partial differential equations
for the transition probabilities of certain infinite-dimensional diffusions.
