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We construct a module action of the affine nil-Hecke algebra $\\hat{H}_*^{S^1 \\times T}(LG/T)$ on the $S^1 \\times T$-equivariant quantum cohomology of $M$, $QH^*_{S^1 \\times T}(M).$ Our construction generalizes the theory of shift operators for Hamiltonian torus actions [OP,LJ]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. 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