Quantum cohomology of partial flag manifolds Is there a place in the literature where the quantum differential
equation (or even just quantum cohomology algebra) 
of partial flag manifolds $G/P$ is computed for arbitrary semi-simple $G$ and 
arbitrary parabolic $P$? I actually think that I know one way 
to formulate (and prove) the answer but
I was sure that this was well-known and to my surprise I couldn't find the reference
for the general case (the case when $P$ is a Borel subgroup is well-known and there is
a lot of literature for other parabolics in the case when $G$ is a classical group but again
I couldn't find a treatment of the general case).
For the quantum cohomology algebra many papers mention a result of Peterson (which I think coincides with what I want when one takes the appropriate limit going from quantum $D$-module to quantum cohomology algebra) which describes it, but I was unable to find a published proof of this result. Is it written anywhere?
 A: Edited in light of clarification made by OP in comments to his question:
Yes, the result you want is proved by Lam and Shimozono; it is Theorem 10.16 of their paper arXiv:0705.1386.  Their theorem (which is followed by a proof) identifies a localization of $QH^T(G/P)$ with a  localization of a quotient of the torus equivariant homology of the affine Grassmannian; specialization gives the earlier non-equivariant unpublished result of Peterson.
The Lam/Shimozono result depends on an earlier calculation (in arXiv:math/0501213) by Mihalcea of the equivariant quantum product of a Schubert class by a divisor class; this rule should already suffice to determine the QDE.
arxiv:1007.1683 by Leung and Li is the state of the art in relations between $QH(G/P)$ and $QH(G/B)$, as far as I am aware.  See in particular Theorem 1.4 (which however restricts to the case P/B equal to a flag variety).
A: 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.
