Linear Structure on a Finite Hecke Category in Type A

Abstract For the group $\textrm{GL}_{n}$, we construct an action of the equivariant derived category of coherent sheaves on the Grothendieck–Springer resolution on a certain subcategory of a finite monodromic Hecke category. We use this to construct a partial categorification of the projection from the extended affine to the finite Hecke algebra of $\textrm{GL}_{n}$. As a crucial intermediate step, we compute the exterior powers, with respect to the perversely truncated multiplicative convolution, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup fixing a line in the defining $n$-dimensional representation of $\textrm{GL}_{n}$.