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?

  • $\begingroup$ Without having expertise in the subject, I'm aware that most of the literature places restrictions on the nature of P and is often focused just on type A. Besides this, Dale Peterson himself is somewhat reclusive and not inclined to publish even his most influential ideas. But quite a few people listing his spring 1997 MIT lectures as a reference must have copies of the notes, even if they are not available online. Probably Konstanze Rietsch at Kings College London as well as people closer to MIT should be asked (such as Bert Kostant). $\endgroup$ Nov 2 '10 at 22:07
  • $\begingroup$ I think I know how to tackle the general case in the spirit similar to my paper arxiv.org/abs/math/0401409 but it would be nice to know whether it is formally new or not. As far as I know both Rietsch and Kostant work with Peterson's answer but they don't know the proof. If Peterson indeed has a proof I would be curious to know what it is... $\endgroup$ Nov 2 '10 at 22:22
  • $\begingroup$ Can you state the result of Peterson more precisely? I'm still somewhat confused about exactly what you're asking for. $\endgroup$ Nov 3 '10 at 13:03
  • $\begingroup$ Well, any kind of description of the quantum cohomology ring (or better the equivariant quantum cohomology ring) would suit me. One such description (at least in the non-eqivariant setting) is indeed due to Peterson and it is stated in many places (e.g. in the papers by Rietsch and Kostant) but I couldn't find a proof of it in the general case. $\endgroup$ Nov 3 '10 at 13:54
  • $\begingroup$ I edited my answer; have you looked at Lam and Shimozono's paper? $\endgroup$ Nov 3 '10 at 14:04

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).

  • $\begingroup$ Thanks, I didn't know about this paper. However, it seems that it doesn't deal with the most general case either: there is an assumption that the Levi group is of type A. $\endgroup$ Nov 2 '10 at 21:29

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.

  • $\begingroup$ The paper by Rietsch is only about type A. Furthemore she doesn't prove Peterson's result in this case -- she cites it in order to do some further things. Peterson's variety is actually not really needed if you want to do equivariant quantum cohomology (which is actually the thing I am interested in). $\endgroup$ Nov 2 '10 at 21:22
  • $\begingroup$ I agree that her proof is only for type A, but Rietsch does prove Peterson's theorem--- the proof starts right after Remark 4.3. It's a shame Peterson's proof was published. Also, sorry I did not realize you were asking about the equivariant case. $\endgroup$
    – Erik Insko
    Nov 4 '10 at 0:34
  • $\begingroup$ Well, she deduces it from previous results of Kim and Ciocan-Fontanine, which as far as I understand are specific for type A (I don't really care in what form the quantum cohomology is computed -- somehow I believe that it shouldn't be difficult to go from one presentation to another. The really interesting thing is to say something about quantum multiplication. If I understand correctly, the paper by Rietsch doesn't deal with that -- it relies on other things. $\endgroup$ Nov 4 '10 at 22:48

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