  We study the algebraic aspects of equivariant quantum cohomology algebra of
the flag manifold. We introduce and study the quantum double Schubert
polynomials, which are the Lascoux-Schutzenberger type representatives of the
equivariant quantum cohomology classes. Our approach is based on the quantum
Cauchy identity. We define also quantum Schubert polynomials as the
Gram-Schmidt orthogonalization of some set of monomials with respect to the
scalar product, defined by the Grothendieck residue. Using quantum Cauchy
identity, we prove that quantum Schubert polynomials are the specialization of
quantum double Schubert polynomials with second set of variables equals to
zero, and as a corollary obtain a simple formula for the quantum Schubert
polynomials. We also prove the higher genus analog of Vafa-Intriligator's
formula for the flag manifolds and study the quantum residues generating
function. We introduce the extended Ehresman-Bruhat order on the symmetric
group and formulate the equivariant quantum Pieri rule.
