Quaternionic Satake equivalence

Abstract We establish a derived geometric Satake equivalence for the quaternionic general linear group ${\textrm{GL}}_{n}({\mathbb H})$ . By applying the real–symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ . We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ . We show that the stalks are given by the Kostka–Foulkes polynomials for ${\textrm{GL}}_n$ but with all degrees doubled.