  We prove a conjecture of Frenkel, Gaitsgory, Kazhdan and Vilonen, related to
Fourier coefficients of spherical perverse sheaves on the affine Grassmannian
associated to a a split reductive group. Our proof is an extension of the proof
given by the first author in the case of GL(n) (see math/9801109); it relies on
the study of certain resolutions of Schubert varieties in the affine
Grassmannian, built from the so-called minuscule or quasi-minuscule cases.

Algebraic Geometry