Kazhdan and Lusztig identified the affine Hecke algebra with equivariant K-theory of Steinberg variety of the dual group. Bezrukavnikov categorified this identification, and in particular, gave a geometric description of the Kazhdan--Lusztig canonical basis. It is given by classes of simple perverse modules over the non-commutative resolution. We propose an analogous construction for a different situation: resolution of affine Schubert variety in type A. We construct the non-commutative resolution and describe the corresponding basis in terms of the quantum loop group action. We emphasize a relation to categorical Howe duality and K-theoretic Satake.

Based on a work in progress, joint with E. Bodish and V. Krylov.