  We make precise conjectures relating the genus zero Gromov-Witten theory of a
nonabelian GIT quotient X//G to that of the associated abelian quotient X//T by
a maximal torus T in G.These conjectures imply in particular closed formulas
for the J-functions (that is, the generating functions for 1-point
Gromov-Witten invariants) of all generalized flag manifolds of classical A, B,
C and D types. These formulas are proved in the second part of the paper.
