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Among many other nice results, the paper "Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties" by Konstanze Rietsch contains a proof of Peterson's result. It's available at arXiv:math/0112024. The result appears as Theorem 4.2.

I believe Peterson's theorem says that if one takes the opposite Schubert cell $B_{-} w_P B/B$ and intersects that with what is now called the Peterson variety, then the coordinate ring of that space is the quantum cohomology of $G/P$.

Section 2 of Harada and Tymoczko's paper "A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties" has a concise description of the Peterson variety. This paper is available on the arxiv at arXiv:0908.3517.

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Among many other nice results, the paper "Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties" by Konstanze Rietsch contains a proof of Peterson's result. It's available at arXiv:math/0112024. The result appears as Theorem 4.2.