  Using Quot schemes and a localization theorem we study Gromov-Witten
invariants for partial flag varieties. The strategy is to extend A. Bertram's
result of Gromov-Witten invariants for special Schubert varieties of
Grassmannians to the case of partial flag varieties. To do so a Grothendieck's
Quot scheme is generalized for flags and proven to be an irreducible, rational,
smooth, projective variety following Str\o mme \cite{St}.
