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A pairing between “Schubert” $H_*(\operatorname{Gr})$ and “Langlands” $H^*(\operatorname{Gr})$

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. …