Noncommutative resolutions of affine Schubert varieties in type A and canonical bases

Given a resolution ${\widetilde{\mathrm{G}\mathrm{r}}}^{\underset{‾}{\lambda }}\to {\overline{\mathrm{G}\mathrm{r}}}^{\lambda }$ of an affine Schubert variety for $G{L}_n$, we define its noncommutative version – a sheaf of algebras on ${\overline{\mathrm{G}\mathrm{r}}}^{\lambda }$, derived equivalent to ${\widetilde{\mathrm{G}\mathrm{r}}}^{\underset{‾}{\lambda }}$ as well as its Steinberg versions in both zero and positive characteristics. This, in particular, allows us to define the perversely-exotic t-structure on the derived category of equivariant coherent sheaves on ${\widetilde{\mathrm{G}\mathrm{r}}}^{\underset{‾}{\lambda }}$, analogously to Bezrukavnikov–Mirković in the case of Springer resolution. We study the basis of classes of irreducible objects in the equivariant K-theory, and explicitly identify it with the (parabolic) Kazhdan–Lusztig canonical basis in a certain cell quotient. This allows us to relate it to the canonical basis for the quantum affine group. In the course of the proof, we establish some properties of coherent-constructible equivalences.