Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X  . As an application, we reduce the study of invariant integrally closed ideals of V×XV×X to that of some smaller variety in the case of X=Mn,mX=Mn,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237–262] for two special infinite series.