Categorical geometric symmetric Howe duality

Abstract

We provide a natural geometric setting for symmetric Howe duality. This is realized as a (loop) $$\mathfrak {sl}_n$$ action on derived categories of coherent sheaves on certain varieties arising in the geometry of the Beilinson–Drinfeld Grassmannian. The main construction parallels our earlier work on categorical $$\mathfrak {sl}_n$$ actions and skew Howe duality. In that case the varieties involved arose in the geometry of the affine Grassmannian. We discuss some relationships between the two actions.