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Let $\operatorname{Gr}$ be an affine Grassmanian of some complex semisimple group $G$. Of course, there is a well-known description of $H^*(\operatorname{Gr})$ in terms of Langlands dual Lie algebra.

On the other hand, one has the base of usual Schubert cycles in $H_*(\operatorname{Gr})$.

The two descriptions seem to be somewhat unrelated; so the following question is quite natural: is there any explicit description of the corresponding pairing?

Thanks in advance!

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  • $\begingroup$ There's a lot to say and I don't know all of it, so I would point you to the book arxiv.org/abs/1301.3569 $\endgroup$
    – Allen Knutson
    Commented Oct 9, 2022 at 21:52
  • $\begingroup$ @AllenKnutson, Dear Allen, thank you for the answer! I can make my question much more precise. In their famous work, Yun and Zhu have shown that the integral cohomology algebra of $Gr$ contains divided powers (namely, it is an enveloping algebra with divided powers). Can these divided powers (namely, their denominators) be seen from the purely combinatorial reasons? (As described in the post.) $\endgroup$
    – Vanya Karpov
    Commented Oct 10, 2022 at 13:23

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