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Seidel-Smith and Manolescu constructed knot homology theories usingsymplectic fibrations whose total spaces were certain varieties of matrices.These knot homology theories were associated to $SL(n) $ and tensor products ofthe standard and dual representations. In this paper, we place their geometricsetups in a natural, general framework. For any complex reductive group and anysequence of minuscule dominant weights, we construct a fibration of affinevarieties over a configuration space. The middle cohomology of these varietiesis isomorphic to the space of invariants in the corresponding tensor product ofrepresentations. Our construction uses the Beilinson-Drinfeld Grassmannian andthe geometric Satake correspondence.

The Beilinson-Drinfeld Grassmannian and symplectic knot homology