We observe that the small quantum product of the generalized flag manifold GIB is a product operation star on H* (G/B) circle times R[q(1), ..., q(l)] uniquely determined by the facts that it is a deformation of the cup product on H* (G/B); it is commutative, associative, and graded with respect to deg(q(i)) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H* (G/B) circle times R[q(1), ..., q(l)], star) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B: the quantum Chevalley formula of D. Peterson (see also Fulton and Woodward) and the quantization by standard monomials" formula of Fomin, Gelfand, and Postnikov for G = SL(n, C). The main idea of the proofs is the same as in Amarzaya-Guest: from the quantum D-module of G/B one can decode all information about the quantum cohomology of this space.
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